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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

Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the region.

y = sin x ,   y = 2 x / π ,   x 0

To determine

To sketch:

The region enclosed by the given curve and find the area of the region.

Explanation

1) Concept:

i. The intersection of two curves is obtained by solving the simultaneous equation of the curves.

ii. Identify the top and the bottom boundaries of region.

iii. Formula-

The area of the typical rectangle is

yT-yBx, where yT = top boundary curve and yB= bottom boundary curve.

The total area is

A= limni=1nyT-yBx = ab(yT-yB)  dx

2) Given:

y=2xπ and y=sinx,  x0

3) Calculation:

The given curves are y=2xπ and y=sinx,  x0

First, find the intersection of these two curves by solving the simultaneous equation.

2xπ=sinx 

That is

2xπ-sinx=0

x=0,  π2

The graph of the region enclosed by the given curve is

Find the area by integrating with respect to x(since, we found the top curve and the bottom curve from the graph and the restriction on x is given, so it is easy to find the area by integrating with the respective x)

From graph, the top and the bottom curves are

yT=sinx  and yB=2xπ

The graph with the typical rectangle is

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