The Portion Of The Curve { Y = 4x $}$ Between { X = A $}$ And { X = B $}$ Is Rotated 2 Radians About The X-axis. The Volume, { V $} , O F T H E S O L I D G E N E R A T E D I S B E S T R E P R E S E N T E D A S : A . \[ , Of The Solid Generated Is Best Represented As:A. \[ , O F T H Eso L I D G E N Er A T E D I S B Es T Re P Rese N T E D A S : A . \[ V = 4 \int \pi

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The Portion of the Curve y=4xy = 4x Rotated About the X-Axis: Calculating the Volume of the Solid Generated

In mathematics, the process of rotating a curve about an axis is a fundamental concept in calculus, particularly in the study of volumes of solids. When a portion of a curve is rotated about the x-axis, it generates a solid with a specific volume. In this article, we will explore the process of calculating the volume of the solid generated by rotating the portion of the curve y=4xy = 4x between x=ax = a and x=bx = b about the x-axis.

The problem involves rotating the portion of the curve y=4xy = 4x between x=ax = a and x=bx = b about the x-axis by 22 radians. This means that the curve will be rotated in a circular motion around the x-axis, generating a solid with a specific volume. The volume of the solid generated is represented by the variable VV.

To calculate the volume of the solid generated, we will use the method of disks. This method involves dividing the solid into thin disks, each with a radius equal to the distance from the x-axis to the curve at a given point. The volume of each disk is then calculated and summed up to find the total volume of the solid.

The volume of each disk is given by the formula:

V=πr2ΔxV = \pi r^2 \Delta x

where rr is the radius of the disk and Δx\Delta x is the thickness of the disk.

In this case, the radius of each disk is equal to the distance from the x-axis to the curve at a given point, which is given by the equation y=4xy = 4x. Therefore, the radius of each disk is 4x4x.

The thickness of each disk is Δx\Delta x, which is the difference between the x-coordinates of two consecutive points on the curve.

Substituting these values into the formula, we get:

V=π(4x)2ΔxV = \pi (4x)^2 \Delta x

Simplifying the expression, we get:

V=16πx2ΔxV = 16 \pi x^2 \Delta x

To find the total volume of the solid generated, we need to integrate the expression for the volume of each disk with respect to xx.

The limits of integration are aa and bb, which represent the x-coordinates of the two points on the curve that define the portion of the curve to be rotated.

Therefore, the integral is:

V=16π∫abx2ΔxV = 16 \pi \int_a^b x^2 \Delta x

Evaluating the integral, we get:

V=16Ï€[x33]abV = 16 \pi \left[\frac{x^3}{3}\right]_a^b

Simplifying the expression, we get:

V=16π3(b3−a3)V = \frac{16 \pi}{3} (b^3 - a^3)

In conclusion, the volume of the solid generated by rotating the portion of the curve y=4xy = 4x between x=ax = a and x=bx = b about the x-axis is given by the expression:

V=16π3(b3−a3)V = \frac{16 \pi}{3} (b^3 - a^3)

This expression represents the volume of the solid generated by rotating the portion of the curve about the x-axis.

The final answer is:

V = \frac{16 \pi}{3} (b^3 - a^3)$<br/> **The Portion of the Curve $y = 4x$ Rotated About the X-Axis: Calculating the Volume of the Solid Generated**

Q: What is the method of disks?

A: The method of disks is a technique used to calculate the volume of a solid generated by rotating a curve about an axis. It involves dividing the solid into thin disks, each with a radius equal to the distance from the axis to the curve at a given point. The volume of each disk is then calculated and summed up to find the total volume of the solid.

Q: How do I calculate the volume of each disk?

A: To calculate the volume of each disk, you need to use the formula:

V=πr2Δx</span></p><p>where<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotationencoding="application/x−tex">r</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">r</span></span></span></span>istheradiusofthediskand<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="normal">Δ</mi><mi>x</mi></mrow><annotationencoding="application/x−tex">Δx</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6833em;"></span><spanclass="mord">Δ</span><spanclass="mordmathnormal">x</span></span></span></span>isthethicknessofthedisk.</p><h2><strong>Q:Whatistheradiusofeachdiskinthisproblem?</strong></h2><p>A:Inthisproblem,theradiusofeachdiskisequaltothedistancefromthex−axistothecurveatagivenpoint,whichisgivenbytheequation<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mn>4</mn><mi>x</mi></mrow><annotationencoding="application/x−tex">y=4x</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.625em;vertical−align:−0.1944em;"></span><spanclass="mordmathnormal"style="margin−right:0.03588em;">y</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">4</span><spanclass="mordmathnormal">x</span></span></span></span>.Therefore,theradiusofeachdiskis<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn><mi>x</mi></mrow><annotationencoding="application/x−tex">4x</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">4</span><spanclass="mordmathnormal">x</span></span></span></span>.</p><h2><strong>Q:HowdoIintegratetofindthetotalvolumeofthesolidgenerated?</strong></h2><p>A:Tofindthetotalvolumeofthesolidgenerated,youneedtointegratetheexpressionforthevolumeofeachdiskwithrespectto<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotationencoding="application/x−tex">x</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal">x</span></span></span></span>.Thelimitsofintegrationare<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotationencoding="application/x−tex">a</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal">a</span></span></span></span>and<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi></mrow><annotationencoding="application/x−tex">b</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mordmathnormal">b</span></span></span></span>,whichrepresentthex−coordinatesofthetwopointsonthecurvethatdefinetheportionofthecurvetoberotated.</p><h2><strong>Q:Whatisthefinalexpressionforthevolumeofthesolidgenerated?</strong></h2><p>A:Thefinalexpressionforthevolumeofthesolidgeneratedis:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>V</mi><mo>=</mo><mfrac><mrow><mn>16</mn><mi>π</mi></mrow><mn>3</mn></mfrac><mostretchy="false">(</mo><msup><mi>b</mi><mn>3</mn></msup><mo>−</mo><msup><mi>a</mi><mn>3</mn></msup><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">V=16π3(b3−a3)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6833em;"></span><spanclass="mordmathnormal"style="margin−right:0.22222em;">V</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:2.0074em;vertical−align:−0.686em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.3214em;"><spanstyle="top:−2.314em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord">3</span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.677em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord">16</span><spanclass="mordmathnormal"style="margin−right:0.03588em;">π</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.686em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span><spanclass="mopen">(</span><spanclass="mord"><spanclass="mordmathnormal">b</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8641em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">3</span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1.1141em;vertical−align:−0.25em;"></span><spanclass="mord"><spanclass="mordmathnormal">a</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8641em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">3</span></span></span></span></span></span></span></span><spanclass="mclose">)</span></span></span></span></span></p><h2><strong>Q:Whataretheassumptionsmadeinthisproblem?</strong></h2><p>A:Theassumptionsmadeinthisproblemare:</p><ul><li>Thecurveisrotatedaboutthex−axis.</li><li>Therotationisby<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn></mrow><annotationencoding="application/x−tex">2</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">2</span></span></span></span>radians.</li><li>Theportionofthecurvetoberotatedisbetween<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mi>a</mi></mrow><annotationencoding="application/x−tex">x=a</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal">a</span></span></span></span>and<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mi>b</mi></mrow><annotationencoding="application/x−tex">x=b</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mordmathnormal">b</span></span></span></span>.</li></ul><h2><strong>Q:Whatarethelimitationsofthisproblem?</strong></h2><p>A:Thelimitationsofthisproblemare:</p><ul><li>Theproblemassumesthatthecurveisrotatedaboutthex−axis,whichmaynotbethecaseinallsituations.</li><li>Theproblemassumesthattherotationisby<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn></mrow><annotationencoding="application/x−tex">2</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">2</span></span></span></span>radians,whichmaynotbethecaseinallsituations.</li><li>Theproblemassumesthattheportionofthecurvetoberotatedisbetween<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mi>a</mi></mrow><annotationencoding="application/x−tex">x=a</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal">a</span></span></span></span>and<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mi>b</mi></mrow><annotationencoding="application/x−tex">x=b</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mordmathnormal">b</span></span></span></span>,whichmaynotbethecaseinallsituations.</li></ul><h2><strong>Q:HowcanIapplythisproblemtoreal−worldsituations?</strong></h2><p>A:Thisproblemcanbeappliedtoreal−worldsituationssuchas:</p><ul><li>Calculatingthevolumeofasolidgeneratedbyrotatingacurveaboutanaxis.</li><li>Designingacontaineroravesselwithaspecificvolume.</li><li>Calculatingthevolumeofasolidgeneratedbyrotatingacurveaboutanaxisinathree−dimensionalspace.</li></ul><p>Inconclusion,theproblemofcalculatingthevolumeofthesolidgeneratedbyrotatingtheportionofthecurve<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mn>4</mn><mi>x</mi></mrow><annotationencoding="application/x−tex">y=4x</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.625em;vertical−align:−0.1944em;"></span><spanclass="mordmathnormal"style="margin−right:0.03588em;">y</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">4</span><spanclass="mordmathnormal">x</span></span></span></span>between<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mi>a</mi></mrow><annotationencoding="application/x−tex">x=a</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal">a</span></span></span></span>and<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mi>b</mi></mrow><annotationencoding="application/x−tex">x=b</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mordmathnormal">b</span></span></span></span>aboutthex−axisisaclassicproblemincalculus.Themethodofdisksisapowerfultechniqueusedtosolvethisproblem,andthefinalexpressionforthevolumeofthesolidgeneratedis:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>V</mi><mo>=</mo><mfrac><mrow><mn>16</mn><mi>π</mi></mrow><mn>3</mn></mfrac><mostretchy="false">(</mo><msup><mi>b</mi><mn>3</mn></msup><mo>−</mo><msup><mi>a</mi><mn>3</mn></msup><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">V=16π3(b3−a3)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6833em;"></span><spanclass="mordmathnormal"style="margin−right:0.22222em;">V</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:2.0074em;vertical−align:−0.686em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.3214em;"><spanstyle="top:−2.314em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord">3</span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.677em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord">16</span><spanclass="mordmathnormal"style="margin−right:0.03588em;">π</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.686em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span><spanclass="mopen">(</span><spanclass="mord"><spanclass="mordmathnormal">b</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8641em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">3</span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1.1141em;vertical−align:−0.25em;"></span><spanclass="mord"><spanclass="mordmathnormal">a</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8641em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">3</span></span></span></span></span></span></span></span><spanclass="mclose">)</span></span></span></span></span></p><p>Thisproblemcanbeappliedtoreal−worldsituationssuchasdesigningacontaineroravesselwithaspecificvolume,andcalculatingthevolumeofasolidgeneratedbyrotatingacurveaboutanaxisinathree−dimensionalspace.</p>V = \pi r^2 \Delta x </span></p> <p>where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span> is the radius of the disk and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\Delta x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Δ</span><span class="mord mathnormal">x</span></span></span></span> is the thickness of the disk.</p> <h2><strong>Q: What is the radius of each disk in this problem?</strong></h2> <p>A: In this problem, the radius of each disk is equal to the distance from the x-axis to the curve at a given point, which is given by the equation <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mn>4</mn><mi>x</mi></mrow><annotation encoding="application/x-tex">y = 4x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">4</span><span class="mord mathnormal">x</span></span></span></span>. Therefore, the radius of each disk is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn><mi>x</mi></mrow><annotation encoding="application/x-tex">4x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">4</span><span class="mord mathnormal">x</span></span></span></span>.</p> <h2><strong>Q: How do I integrate to find the total volume of the solid generated?</strong></h2> <p>A: To find the total volume of the solid generated, you need to integrate the expression for the volume of each disk with respect to <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span>. The limits of integration are <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span></span></span></span>, which represent the x-coordinates of the two points on the curve that define the portion of the curve to be rotated.</p> <h2><strong>Q: What is the final expression for the volume of the solid generated?</strong></h2> <p>A: The final expression for the volume of the solid generated is:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>V</mi><mo>=</mo><mfrac><mrow><mn>16</mn><mi>π</mi></mrow><mn>3</mn></mfrac><mo stretchy="false">(</mo><msup><mi>b</mi><mn>3</mn></msup><mo>−</mo><msup><mi>a</mi><mn>3</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V = \frac{16 \pi}{3} (b^3 - a^3) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">16</span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p> <h2><strong>Q: What are the assumptions made in this problem?</strong></h2> <p>A: The assumptions made in this problem are:</p> <ul> <li>The curve is rotated about the x-axis.</li> <li>The rotation is by <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span> radians.</li> <li>The portion of the curve to be rotated is between <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">x = a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">x = b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span></span></span></span>.</li> </ul> <h2><strong>Q: What are the limitations of this problem?</strong></h2> <p>A: The limitations of this problem are:</p> <ul> <li>The problem assumes that the curve is rotated about the x-axis, which may not be the case in all situations.</li> <li>The problem assumes that the rotation is by <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span> radians, which may not be the case in all situations.</li> <li>The problem assumes that the portion of the curve to be rotated is between <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">x = a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">x = b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span></span></span></span>, which may not be the case in all situations.</li> </ul> <h2><strong>Q: How can I apply this problem to real-world situations?</strong></h2> <p>A: This problem can be applied to real-world situations such as:</p> <ul> <li>Calculating the volume of a solid generated by rotating a curve about an axis.</li> <li>Designing a container or a vessel with a specific volume.</li> <li>Calculating the volume of a solid generated by rotating a curve about an axis in a three-dimensional space.</li> </ul> <p>In conclusion, the problem of calculating the volume of the solid generated by rotating the portion of the curve <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mn>4</mn><mi>x</mi></mrow><annotation encoding="application/x-tex">y = 4x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">4</span><span class="mord mathnormal">x</span></span></span></span> between <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">x = a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">x = b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span></span></span></span> about the x-axis is a classic problem in calculus. The method of disks is a powerful technique used to solve this problem, and the final expression for the volume of the solid generated is:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>V</mi><mo>=</mo><mfrac><mrow><mn>16</mn><mi>π</mi></mrow><mn>3</mn></mfrac><mo stretchy="false">(</mo><msup><mi>b</mi><mn>3</mn></msup><mo>−</mo><msup><mi>a</mi><mn>3</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V = \frac{16 \pi}{3} (b^3 - a^3) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">16</span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p> <p>This problem can be applied to real-world situations such as designing a container or a vessel with a specific volume, and calculating the volume of a solid generated by rotating a curve about an axis in a three-dimensional space.</p>