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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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Chapter
Section
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 = x3, y = x

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=x3 and y=x.

Calculation:

Find the intersection points of the curves by equating the curves as shown below:

f(x)=g(x) (1)

Here, the top curve function is f(x) and the bottom curve function is g(x).

Substitute x for f(x) and x3 for g(x) in Equation (1).

x=x3x3x=0x(x21)=0 (2)

Solve Equation (2).

x=0x21=0x2=1x=±1

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

  • Draw the graph for the function y=x3 by substituting different values for x.
  • Similarly in the same graph plot for the function y=x by substituting different values for x.
  • Shade the region lies between the intersecting points of the curves.

The region enclosed by the curves y=x3 and y=x is shown in Figure 1.

Refer to Figure 1.

The intersecting points has the limit value of x=1 and x=1.

The curves are bounded by the top and bottom curve. Hence, 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 (3)

Here, the lower limit is a and the upper limit is b

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