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ERJAEE, G. (20)
Lecture 14
Department
MathematicsCourse Code
MATH 2BProfessor
ERJAEE, G.Lecture
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02/06/2019
MATH 2B - Lecture 14 - Trigonometric Substitution
We begin by considering the area of a quarter circle with radius a:
● First we know the curve representing a circle is
● Then we have
● Take the square-root on both sides and only keep the
positive part, we have:
(range for the
quarter)
● So the area of a quarter circle can be written as
● Now, we will learn how to use trig. Sub to solve the integral
Using trigonometric substitution
● We first pick up a point on the curve and denote it as (x,y)
● Connecting this point to the center, we obtain a right-angle triangle
○ The long side is the radius a, two legs are x,y
○ Due to the property of right-angle trigonometry, we have:
→
Now we go back to the integral
From the right-angle triangle we know that
and
So we have:
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