This

**preview**shows half of the first page. to view the full**1 pages of the document.**8.1 Challenging Integral Applications

Arc Length

Question #1 (Easy): Length of the Curve Using the Arc Length Formula

Strategy:

Arc length formula in Leibniz notation is:

This requires taking the derivative of the given function with respect to , then plugging into the

integral over the given interval . Then solve using all integral techniques you have learned so far.

Sample Question:

Using the arc length formula, find the length of the curve. Check your answer by noting that the curve is

part of a circle.

,

Solution:

The arc length formula requires the derivative of y with respect to x, meaning

. Since

. Thus:

. Then, the arc of the length is

. This we

can solve by trigonometric substitution. Using sine substitution based on the identity

. Let

. Here , so , then .

.

Whenever substitutions are made for definite integral, the interval needs to be changed to match the

substitution. So when , ; and when ,

. Then:

.

Therefore the length of the curve over the interval is .

Now comparing to the circle’s circumference, from the equation the radius is , and the interval

represents a quarter of the circle, thus . Dividing by gives .

So looking at it either way, the answers match!

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