   Chapter 5.5, Problem 64E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Evaluate the definite integral. ∫ 0 a x a 2 − x 2   d x

To determine

To evaluate: The definite integral.

Explanation

Given:

The definite integral function is 0axa2x2dx.

The region lies between x=0 and x=a.

Calculation:

Consider u=a2x2 (1)

Differentiate both sides of the Equation (1).

du=2xdxxdx=12du

Calculate the lower limit value of u using Equation (1).

Substitute 0for x in Equation (1).

u=a2(0)2=a2

Calculate the upper limit value of u using Equation (1).

Substitute a for x in Equation (1).

u=a2(a)2=0

The definite integral function is,

0axa2x2dx (2)

Apply lower and upper limits for u in Equation (2).

Substitute u for (a2x2) and (12du) for (xdx) in Equation (2).

0axa2x2dx=a20u(12du)=12a20udu=12a20u12du (3)

Integrate Equation (3)

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