   Chapter 11.10, Problem 60E

Chapter
Section
Textbook Problem

Use series to approximate the definite integral to within the indicated accuracy. ∫ 0 0.5 x 2 e − x 2   d x     ( | error | < 0.001 )

To determine

To approximate:

The definite integral to with in the indicated accuracy.

Explanation

1) Concept:

Use the Maclaurin series to evaluate the integral,

ex=n=0xnn! , for all x

2) Given:

00.5 x2e-x2 dx , (error<0.001)

3) Calculation:

The Maclaurin series expansion is ex=n=0xnn! , for all x

Therefore,

x2e-x2=x2n=0-1n(x2)nn! , for all x

(Replacing, x by -x2)

x2e-x2=n=0-1nx2n+2 n! , for all x

Integrating,

x2e-x2dx=n=0-1nx2n+2 n! dx

x2e-x2dx=n=0-1nx2n+2 n!dx

By the power rule of integration,

=n=0-1nx2n+3 n!2n+3+C

=x33-x51!·5+x72!·7-x93!·9+

Therefore,

00

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