   Chapter 7.3, Problem 6E

Chapter
Section
Textbook Problem

# Evaluate the integral. ∫ 0 3 x 36 − x 2   d x

To determine

To evaluate: The given integral 03x36x2dx.

Explanation

Integration involving terms of the form a2x2 can be simplified by using the trigonometric substitution x=asinθ.

Formula used:

The identity, cos2x=1sin2x

Given:

The integral, 03x36x2dx

Calculation:

The integral can be written as:

03x36x2dx=03x62x2dx

Substitute for x as x=6sinθ. Take the derivative of the substitution term:

x=6sinθdx=6cosθdθ

Here, π2θπ2. The limits of integration will change as:

x0sinθ=0θ0andx36sinθ=3sinθ=12θπ6

Substitute for x and dx in the given integral to get:

03x36x2dx=0π66sinθ6

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