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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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Section
BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
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 = (x − 2)2, y = x

To determine

To draw: the region enclosed by given curve.

The area of the region enclosed by the curves.

Explanation

Given information:

The two curves has a function of y=x and y=(x2)2.

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 (x2)2 for g(x) in Equation (1).

x=(x2)2x=x24x+4x24xx+4=0x25x+4=0 (2)

Solve Equation (2).

x25x+4=0x21x4x+4=0x(x1)4(x1)x=1and4

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

  • Draw the graph for the function y=(x2)2 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 x=1 and x=4.

The region enclosed by the curves y=(x2)2 and y=x is shown in Figure 1.

Decide whether the integration can be done with respect to x or y is given below:

  • If the region of curves is bound by the top and bottom curve then the integration can be done with respect to x.
  • If the region of curves is bound by the right and left curve then the integration can be done with respect to y.

Therefore, the integration can be done with respect to x.

Refer to Figure (1).

Draw a typical approximate rectangle with a width (Δx) and height (h).

Find the height of the typical approximating rectangle using the relation:

h=yTyB (3)

Here, the top curve is yT and the bottom curve is yB.

Substitute x  for yT and (x2)2 for yB in Equation (3)

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