# 8.2 Surface Area of Revolution Question #4 (Medium)

8.2 Challenging Integral Applications

Surface Area of Revolution

Question #4 (Medium): Surface Area From Rotating the Function About the Y-Axis

Strategy

When the function is rotated about the -axis, the radius extends horizontally. When the function is

expressed as , then

. But if the function is given as , then

. The expression from arc length can be evaluated with

just as well

because

. The function does not have to be rearranged,

and the derivative can be taken as it is.

Sample Question

Find the exact surface area obtained by rotating the given function about the -axis.

,

Solution

Since the rotation is about the -axis,

. But since the function is given for

in terms of , without having to rearrange, take the derivate as it is for

, then

. So

. Thus:

.

Since it is in chain rule form, substitution can be used. Let , then , so

.

When , then , and when , then . Then:

Therefore, the surface area obtained by rotating the function about the -axis over the

range is

## Document Summary

Question #4 (medium): surface area from rotating the function about the y-axis. When the function is rotated about the -axis, the radius extends horizontally. When the function is expressed as , then ( But if the function is given as , then. The expression from arc length can be evaluated with just as well because ( The function does not have to be rearranged, and the derivative can be taken as it is. Find the exact surface area obtained by rotating the given function about the -axis. Since the rotation is about the -axis, ( But since the function is given for in terms of , without having to rearrange, take the derivate as it is for. Since it is in chain rule form, substitution can be used.