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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. 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 = 1/x, y = 1/x2, x = 2

To determine

The area of the region enclosed by the two curves.

Explanation

Given information:

The two curves has the function y=1x and y=1x2.

The curve lies at x=2.

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

  • Draw the graph for the function y=1x by substituting different values for x.
  • Similarly plot for the function y=1x2 by substituting different values for x.
  • Represent the line x=2.

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

The ways to find 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, for the given curves the integration can be done with respect to x.

From 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 (1)

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

Substitute 1x for yT and 1x2 for yB in Equation (1).

h=1x1x2

The typical approximating rectangle with dimensions is shown in Figure 2

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