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Chapter 6: Problem 4

Suppose \(g\) is positive and differentiable on \([c, d] .\) The curve \(x=g(y)\) on \([c, d]\) is revolved about the \(y\) -axis. Explain how to find the area of the surface that is generated.

Short Answer

Expert verified
Question: Determine the surface area of the curve x=g(y) on the interval [c, d] when revolved around the y-axis.

Step by step solution

01

Define the function and interval

The function g(y) is given, defined on the interval [c, d].
02

Set up Surface Area of Revolution formula for y-axis

To obtain the Surface Area of Revolution (S) when rotating around the y-axis, use the formula: $$ S = 2\pi \int_{c}^{d} x \sqrt{1 + (\frac{dx}{dy})^2} dy $$ Here, $$x = g(y)$$ and $$\frac{dx}{dy} = g'(y)$$.
03

Substitute x and dx/dy into the formula

Replace x with g(y) and $$\frac{dx}{dy}$$ with g'(y) in the surface area formula: $$ S = 2\pi \int_{c}^{d} g(y) \sqrt{1 + (g'(y))^2} dy $$
04

Evaluate the integral

Lastly, evaluate the integral to find the surface area of the curve x=g(y) revolved around the y-axis: $$ S = 2\pi \int_{c}^{d} g(y) \sqrt{1 + (g'(y))^2} dy $$

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