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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

Evaluate the integral by interpreting it in terms of areas.

5 5 ( x 25 x 2 ) d x

To determine

To evaluate:

The integral

-55x-25-x2dx

by interpreting it in terms of areas.

Explanation

1) Concept:

A definite integral can be interpreted as a net area, that is, a difference of areas;

abf(x)dx=A1-A2

where A1 is the region above x- axis and below the graph of f, and A2 is the region below x- axis and above the graph of f.

2) The property of integral

abfx-gxdx=abf(x)dx-abgx dx

3) Formula:

Area of semi-circle =  A=12(πr2)

4) Given:

-55x-25-x2dx

5) Calculation:

By using property of integral,

-55x-25-x2dx=-55xdx--5525-x2 dx

Now, draw the graph for each part separately.

First, drawgraph of fx= x in the given domain.

The graph of y=x is the line with slope 1 as shown below

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