8.1 Challenging Integral Applications

Arc Length

Question #2 (Medium): Integral Representation of the Length of the Curve

Strategy:

When the function is given as in terms of , take the derivative of in terms of , thus

.

If the function is given as in terms of , no need to rearrange to have written in terms of . take the

derivative of it as it is, thus

, and taking the integral in terms of .

Sample Question:

Set up an integral representing the length of the curve. Then evaluate the exact length of the curve.

,

Solution:

First take the derivative of the given function. Thus,

. Plug into the formula for the

curve length:

. Based on the identity

:

. multiply by

:

Therefore, the length of the curve over the interval

is

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###### Document Summary

Question #2 (medium): integral representation of the length of the curve. When the function is given as in terms of , take the derivative of in terms of , thus. If the function is given as in terms of , no need to rearrange to have written in terms of . take the derivative of it as it is, thus. , and taking the integral in terms of . Set up an integral representing the length of the curve. Then evaluate the exact length of the curve. First take the derivative of the given function. Plug into the formula for the curve length: (

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