Chapter 8.4, Problem 37E

### Calculus of a Single Variable

11th Edition
Ron Larson + 1 other
ISBN: 9781337275361

Chapter
Section

### Calculus of a Single Variable

11th Edition
Ron Larson + 1 other
ISBN: 9781337275361
Textbook Problem

# Converting the Limits of Integration In Exercises 37-42, evaluate the definite integral usingThe given integration limits and (b) the limits obtained by trigonometric substitution. ∫ 2 3 t ( 1 − t 2 ) 3 / 2 d t

To determine

To calculate: The value of the given definite integral 032t2(1t2)32dt.

Explanation

Given:

The given definite integral is âˆ«032t2(1âˆ’t2)32dt.

Calculation:

Because (1âˆ’t2) is of the form (a2âˆ’x2), use the trigonometric substitution t=sinÎ¸.

The derivative of above is;

dt=cosÎ¸dÎ¸

When t=0 the value of Î¸ is;

0=sinÎ¸Î¸=0

When t=32 the value of Î¸ is;

32=sinÎ¸Ï€3Î¸=Ï€3

So, the definite integration can be written as;

âˆ«032t2(1âˆ’t2)32dt=âˆ«0Ï€3sin2

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