Find The Area Of The Surface Generated When The Given Curve Is Revolved About The Given Axis.${ Y = 4x - 7 }$for { \frac{9}{4} \leq X \leq \frac{17}{4} $}$; About The { Y $} − A X I S . ( H I N T : I N T E G R A T E W I T H R E S P E C T T O \[ -axis.(Hint: Integrate With Respect To \[ − A X I S . ( H In T : I N T E G R A T E W I T H Res P Ec Tt O \[

by ADMIN 354 views

Introduction

In mathematics, the surface area of a solid of revolution is the total area of the surface of the solid. When a curve is revolved about an axis, it generates a surface. In this article, we will discuss how to find the surface area of a curve revolved about a given axis. We will use the method of integration to find the surface area.

The Method of Integration

The method of integration is used to find the surface area of a curve revolved about an axis. The formula for the surface area of a curve revolved about the y-axis is given by:

S=2π∫abx1+(dxdy)2 dy{ S = 2\pi \int_{a}^{b} x \sqrt{1 + \left( \frac{dx}{dy} \right)^2} \, dy }

where S is the surface area, x is the function of y, and dx/dy is the derivative of x with respect to y.

The Given Curve

The given curve is:

y=4x−7{ y = 4x - 7 }

This is a linear function of x. To find the surface area, we need to find the derivative of x with respect to y.

Finding the Derivative

To find the derivative of x with respect to y, we can use the chain rule. The derivative of x with respect to y is given by:

dxdy=14{ \frac{dx}{dy} = \frac{1}{4} }

Finding the Surface Area

Now that we have the derivative of x with respect to y, we can find the surface area using the formula:

S=2π∫abx1+(dxdy)2 dy{ S = 2\pi \int_{a}^{b} x \sqrt{1 + \left( \frac{dx}{dy} \right)^2} \, dy }

Substituting the values of x and dx/dy, we get:

S=2π∫94174(y+74)1+(14)2 dy{ S = 2\pi \int_{\frac{9}{4}}^{\frac{17}{4}} \left( \frac{y + 7}{4} \right) \sqrt{1 + \left( \frac{1}{4} \right)^2} \, dy }

Simplifying the expression, we get:

S=2π∫94174(y+74)1716 dy{ S = 2\pi \int_{\frac{9}{4}}^{\frac{17}{4}} \left( \frac{y + 7}{4} \right) \sqrt{\frac{17}{16}} \, dy }

S=178∫94174(y+7) dy{ S = \frac{\sqrt{17}}{8} \int_{\frac{9}{4}}^{\frac{17}{4}} (y + 7) \, dy }

Evaluating the integral, we get:

S=178[y22+7y]94174{ S = \frac{\sqrt{17}}{8} \left[ \frac{y^2}{2} + 7y \right]_{\frac{9}{4}}^{\frac{17}{4}} }

S=178[(28932+1198)−(8132+638)]{ S = \frac{\sqrt{17}}{8} \left[ \left( \frac{289}{32} + \frac{119}{8} \right) - \left( \frac{81}{32} + \frac{63}{8} \right) \right] }

S=178[20832]{ S = \frac{\sqrt{17}}{8} \left[ \frac{208}{32} \right] }

S=178[132]{ S = \frac{\sqrt{17}}{8} \left[ \frac{13}{2} \right] }

S=131716{ S = \frac{13\sqrt{17}}{16} }

Conclusion

In this article, we discussed how to find the surface area of a curve revolved about a given axis. We used the method of integration to find the surface area. The formula for the surface area of a curve revolved about the y-axis is given by:

S=2π∫abx1+(dxdy)2 dy{ S = 2\pi \int_{a}^{b} x \sqrt{1 + \left( \frac{dx}{dy} \right)^2} \, dy }

We applied this formula to the given curve:

y=4x−7{ y = 4x - 7 }

for

94≤x≤174{ \frac{9}{4} \leq x \leq \frac{17}{4} }

about the y-axis.

The surface area of the curve is given by:

S=131716{ S = \frac{13\sqrt{17}}{16} }

This is the final answer.

References

  • [1] "Surface Area of a Solid of Revolution" by Wolfram MathWorld
  • [2] "Surface Area of a Solid of Revolution" by Khan Academy

Mathematical Formulas

  • The formula for the surface area of a curve revolved about the y-axis is given by:

S=2π∫abx1+(dxdy)2 dy{ S = 2\pi \int_{a}^{b} x \sqrt{1 + \left( \frac{dx}{dy} \right)^2} \, dy }

  • The derivative of x with respect to y is given by:

dxdy=14{ \frac{dx}{dy} = \frac{1}{4} }

  • The surface area of the curve is given by:

S=131716{ S = \frac{13\sqrt{17}}{16} }

Mathematical Concepts

  • Surface area of a solid of revolution
  • Method of integration
  • Derivative of a function
  • Solid of revolution

Mathematical Operations

  • Integration
  • Derivative
  • Multiplication
  • Addition
  • Subtraction

Mathematical Functions

  • Surface area function
  • Derivative function
  • Integral function

Mathematical Constants

  • Pi (π)
  • Square root of 17 (√17)

Mathematical Variables

  • x
  • y
  • a
  • b
  • S
    Q&A: Surface Area of a Revolved Curve =============================================

Frequently Asked Questions

Q: What is the surface area of a curve revolved about a given axis? A: The surface area of a curve revolved about a given axis is the total area of the surface of the solid generated by revolving the curve about the axis.

Q: How do I find the surface area of a curve revolved about the y-axis? A: To find the surface area of a curve revolved about the y-axis, you can use the formula:

S=2π∫abx1+(dxdy)2 dy{ S = 2\pi \int_{a}^{b} x \sqrt{1 + \left( \frac{dx}{dy} \right)^2} \, dy }

Q: What is the derivative of x with respect to y? A: The derivative of x with respect to y is given by:

dxdy=14{ \frac{dx}{dy} = \frac{1}{4} }

Q: How do I evaluate the integral in the surface area formula? A: To evaluate the integral in the surface area formula, you can use the following steps:

  1. Substitute the values of x and dx/dy into the formula.
  2. Simplify the expression.
  3. Evaluate the integral using the fundamental theorem of calculus.

Q: What is the surface area of the curve y = 4x - 7 for 9/4 ≤ x ≤ 17/4 about the y-axis? A: The surface area of the curve y = 4x - 7 for 9/4 ≤ x ≤ 17/4 about the y-axis is given by:

S=131716{ S = \frac{13\sqrt{17}}{16} }

Q: Can I use the surface area formula for curves revolved about other axes? A: Yes, you can use the surface area formula for curves revolved about other axes. However, you will need to modify the formula to account for the axis of revolution.

Q: What are some common applications of surface area formulas? A: Surface area formulas have many common applications in mathematics and science, including:

  • Calculating the surface area of a solid of revolution
  • Finding the volume of a solid of revolution
  • Determining the surface area of a curve revolved about a given axis
  • Calculating the surface area of a parametric curve

Q: How do I determine the surface area of a parametric curve? A: To determine the surface area of a parametric curve, you can use the following steps:

  1. Find the parametric equations of the curve.
  2. Calculate the derivative of the curve with respect to the parameter.
  3. Use the surface area formula to calculate the surface area of the curve.

Q: What are some common mistakes to avoid when using surface area formulas? A: Some common mistakes to avoid when using surface area formulas include:

  • Failing to account for the axis of revolution
  • Not simplifying the expression before evaluating the integral
  • Not using the correct formula for the surface area of a curve revolved about a given axis

Q: Can I use surface area formulas to calculate the surface area of a surface of revolution? A: Yes, you can use surface area formulas to calculate the surface area of a surface of revolution. However, you will need to modify the formula to account for the surface of revolution.

Q: What are some common applications of surface area formulas in real-world problems? A: Surface area formulas have many common applications in real-world problems, including:

  • Calculating the surface area of a building or a bridge
  • Determining the surface area of a curve revolved about a given axis
  • Finding the volume of a solid of revolution
  • Calculating the surface area of a parametric curve

Q: How do I determine the surface area of a surface of revolution? A: To determine the surface area of a surface of revolution, you can use the following steps:

  1. Find the parametric equations of the surface of revolution.
  2. Calculate the derivative of the surface of revolution with respect to the parameter.
  3. Use the surface area formula to calculate the surface area of the surface of revolution.

Q: What are some common mistakes to avoid when using surface area formulas to calculate the surface area of a surface of revolution? A: Some common mistakes to avoid when using surface area formulas to calculate the surface area of a surface of revolution include:

  • Failing to account for the surface of revolution
  • Not simplifying the expression before evaluating the integral
  • Not using the correct formula for the surface area of a surface of revolution