Find The Volume Of The Solid Obtained By Rotating The Region Bounded By X = − 3 + Y 2 X = -3 + Y^2 X = − 3 + Y 2 And X = − 2 Y X = -2y X = − 2 Y About The Line X = − 7 X = -7 X = − 7 . Round To The Nearest Thousandth.

Introduction

In mathematics, the process of finding the volume of a solid obtained by rotating a region about a line is a fundamental concept in calculus. This process involves using the method of disks or washers to calculate the volume of the resulting solid. In this article, we will explore how to find the volume of a solid obtained by rotating the region bounded by the equations $x = -3 + y^2$ and $x = -2y$ about the line $x = -7$. We will use the method of disks to calculate the volume and round the result to the nearest thousandth.

Understanding the Problem

To begin, let's understand the problem at hand. We are given two equations, $x = -3 + y^2$ and $x = -2y$, which bound a region in the coordinate plane. We are asked to find the volume of the solid obtained by rotating this region about the line $x = -7$. This means that we will be rotating the region around a vertical line, and the resulting solid will be a three-dimensional shape.

Visualizing the Region

To visualize the region bounded by the two equations, we can start by graphing the two equations on the same coordinate plane. The equation $x = -3 + y^2$ is a parabola that opens to the right, while the equation $x = -2y$ is a line with a negative slope. By graphing these two equations, we can see that they intersect at two points, which bound the region.

Finding the Intersection Points

To find the intersection points of the two equations, we can set them equal to each other and solve for $y$. This gives us the equation $-3 + y^2 = -2y$. Rearranging this equation, we get $y^2 + 2y - 3 = 0$. We can solve this quadratic equation using the quadratic formula, which gives us two solutions for $y$: $y = -3$ and $y = 1$. These are the $y$-coordinates of the intersection points.

Finding the Volume Using the Method of Disks

Now that we have found the intersection points, we can use the method of disks to find the volume of the solid obtained by rotating the region about the line $x = -7$. The method of disks involves integrating the area of the disks formed by rotating the region about the line. In this case, the disks are formed by rotating the region about the line $x = -7$, which is a vertical line.

Calculating the Volume

To calculate the volume, we need to integrate the area of the disks with respect to $y$. The area of each disk is given by the formula $A(y) = \pi (f(y))^2$, where $f(y)$ is the distance from the line $x = -7$ to the curve $x = -3 + y^2$. This distance is given by $f(y) = -3 + y^2 - (-7) = y^2 + 4$.

Setting Up the Integral

Now that we have found the area of each disk, we can set up the integral to calculate the volume. The integral is given by:

$$V = \pi \int_{-3}^{1} (y^2 + 4)^2 dy$$

Evaluating the Integral

To evaluate this integral, we can expand the integrand and then integrate term by term. This gives us:

$$V = \pi \int_{-3}^{1} (y^4 + 8y^2 + 16) dy$$

Integrating Term by Term

Now that we have expanded the integrand, we can integrate term by term. This gives us:

$$V = \pi \left[\frac{y^5}{5} + \frac{8y^3}{3} + 16y\right]_{-3}^{1}$$

Evaluating the Limits

To evaluate the limits, we can substitute the values of $y$ into the expression. This gives us:

$$V = \pi \left[\left(\frac{1}{5} + \frac{8}{3} + 16\right) - \left(\frac{-243}{5} + \frac{-512}{3} + 48\right)\right]$$

Simplifying the Expression

Now that we have evaluated the limits, we can simplify the expression. This gives us:

$$V = \pi \left[\frac{1}{5} + \frac{8}{3} + 16 + \frac{243}{5} + \frac{512}{3} + 48\right]$$

Combining Like Terms

To combine like terms, we can first find a common denominator for the fractions. This gives us:

$$V = \pi \left[\frac{3}{15} + \frac{40}{15} + \frac{240}{15} + \frac{729}{15} + \frac{1600}{15} + \frac{720}{15}\right]$$

Adding the Fractions

Now that we have found a common denominator, we can add the fractions. This gives us:

$$V = \pi \left[\frac{3 + 40 + 240 + 729 + 1600 + 720}{15}\right]$$

Simplifying the Fraction

To simplify the fraction, we can add the numerators. This gives us:

$$V = \pi \left[\frac{3732}{15}\right]$$

Dividing by 15

To divide by 15, we can multiply by the reciprocal of 15. This gives us:

$$V = \pi \left[\frac{3732}{15} \times \frac{1}{15}\right]$$

Multiplying the Numerators

Now that we have multiplied the numerators, we can simplify the fraction. This gives us:

$$V = \pi \left[\frac{3732}{225}\right]$$

Rounding to the Nearest Thousandth

Finally, we can round the result to the nearest thousandth. This gives us:

$$V \approx 16.573$$

Conclusion

In this article, we have used the method of disks to find the volume of a solid obtained by rotating a region about a line. We have graphed the region, found the intersection points, and then used the method of disks to calculate the volume. We have also rounded the result to the nearest thousandth. This process has given us a deeper understanding of the concept of volume and how it can be calculated using calculus.

Q: What is the method of disks?

A: The method of disks is a technique used in calculus to find the volume of a solid obtained by rotating a region about a line. It involves integrating the area of the disks formed by rotating the region about the line.

Q: How do I find the intersection points of two curves?

A: To find the intersection points of two curves, you can set the two equations equal to each other and solve for the variable. This will give you the x-coordinates of the intersection points.

Q: What is the formula for the area of a disk?

A: The formula for the area of a disk is A(y) = π(f(y))^2, where f(y) is the distance from the line to the curve.

Q: How do I set up the integral to calculate the volume?

A: To set up the integral, you need to integrate the area of the disks with respect to y. The integral is given by V = π ∫[a,b] (f(y))^2 dy, where a and b are the limits of integration.

Q: What is the difference between the method of disks and the method of washers?

A: The method of disks and the method of washers are both used to find the volume of a solid obtained by rotating a region about a line. The main difference between the two methods is that the method of disks involves integrating the area of the disks formed by rotating the region about the line, while the method of washers involves integrating the area of the washers formed by rotating the region about the line.

Q: How do I evaluate the integral to calculate the volume?

A: To evaluate the integral, you need to integrate term by term. This involves expanding the integrand and then integrating each term separately.

Q: What is the final answer for the volume of the solid obtained by rotating the region about the line?

A: The final answer for the volume of the solid obtained by rotating the region about the line is approximately 16.573.

Q: Can I use the method of disks to find the volume of any solid obtained by rotating a region about a line?

A: Yes, you can use the method of disks to find the volume of any solid obtained by rotating a region about a line. However, you need to make sure that the region is bounded by two curves and that the line is a vertical line.

Q: What are some common mistakes to avoid when using the method of disks?

A: Some common mistakes to avoid when using the method of disks include:

• Not finding the intersection points of the two curves
• Not setting up the integral correctly
• Not evaluating the integral correctly
• Not rounding the result to the nearest thousandth

Q: How can I practice using the method of disks?

A: You can practice using the method of disks by working through examples and exercises. You can also try using the method of disks to find the volume of different solids obtained by rotating regions about lines.

Q: What are some real-world applications of the method of disks?

A: The method of disks has many real-world applications, including:

• Finding the volume of a tank or a container
• Finding the volume of a solid object
• Finding the volume of a region in space
• Calculating the volume of a solid obtained by rotating a region about a line in engineering and physics.

Q: Can I use the method of disks to find the volume of a solid obtained by rotating a region about a horizontal line?

A: Yes, you can use the method of disks to find the volume of a solid obtained by rotating a region about a horizontal line. However, you need to make sure that the region is bounded by two curves and that the line is a horizontal line.

Q: What are some common difficulties when using the method of disks?

A: Some common difficulties when using the method of disks include:

• Finding the intersection points of the two curves
• Setting up the integral correctly
• Evaluating the integral correctly
• Rounding the result to the nearest thousandth.

Q: How can I overcome these difficulties?

A: You can overcome these difficulties by:

• Practicing using the method of disks
• Working through examples and exercises
• Using online resources and tutorials
• Asking for help from a teacher or a tutor.