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ERJAEE, G. (20)

Lecture 14

Department

MathematicsCourse Code

MATH 2BProfessor

ERJAEE, G.Lecture

14This

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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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