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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

Sketch the region enclosed by the given curves and find its area.

y = cos x, y = 2 – cos x, 0 ≤ x ≤ 2π

To determine

To draw: The region enclosed by the given curves.

The area of the region enclosed by the curves.

Explanation

Given information:

The two curves has a function of y=cosx and y=2cosx.

The upper limit is 2π and the lower limit is 0.

Procedure to sketch the region bounded by the two curves is explained below:

  • Draw the graph for the function y=cosx by substituting different values for x.
  • Similarly in the same graph plot for the function y=2cosx by substituting different values for x.
  • Shade the region lies between x=0 and x=2π.

The region enclosed by the curves y=cosx and y=2cosx is shown in Figure 1.

Refer to Figure 1.

The curves are bounded by the top and bottom curve, so the integration can be done with respect to x.

Find the area of the region bounded by the curves using the relation:

A=ab(f(x)g(x))dx (1)

Here, the top curve is f(x), the bottom curve is g(x), the lower limit is a, and the upper limit is b.

Substitute 2cosx for f(x), cosx for g(x), 0 for a, and 2π for b in Equation (1)

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