MAT136H1 Lecture Notes - Arc Length, Trigonometric Substitution
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8.1 Challenging Integral Applications
Question #1 (Easy): Length of the Curve Using the Arc Length Formula
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.
Using the arc length formula, find the length of the curve. Check your answer by noting that the curve is
part of a circle.
The arc length formula requires the derivative of y with respect to x, meaning
. Then, the arc of the length is
. This we
can solve by trigonometric substitution. Using sine substitution based on the identity
. Here , so , then .
Whenever substitutions are made for definite integral, the interval needs to be changed to match the
substitution. So when , ; and when ,
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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