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**preview**shows half of the first page. to view the full**1 pages of the document.**8.2 Challenging Integral Applications

Surface Area of Revolution

Question #1 (Easy): Finding the Surface Area From Rotating the Function About the X-Axis

Strategy

When the function is rotated about the -axis, then the radius extends vertically up. When the function

is expressed as , then use the formula

. But if the function is given in the

form of , then use

. The part that comes from arc length can be

evaluated with

because based on

. So other than

rearrangement of the function is needed to express the vertically stretching radius, but as for the ds

portion, 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

From comes the integral expression for the surface area from rotating the

function about the x-axis:

. Notice it incorporates the formula for arc length. Thus,

first take the derivative of y:

. Plugging into the SA equation:

. substitution can be used. let , then and

. When

, , and when , . Then:

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

region is

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