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Lecture

MAT136H1 Lecture Notes - Arc Length, Trigonometric Substitution


Department
Mathematics
Course Code
MAT136H1
Professor
all

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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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