   Chapter 15, Problem 49RE

Chapter
Section
Textbook Problem

If D is the region bounded by the curves y = 1 − x2 and y = ex, find the approximate value of the integral ∬ D y 2   d A . (Use a graphing device to estimate the points of intersection of the curves.)

To determine

To find: The approximate the value of the given integral.

Explanation

Given:

The region D is bounded by the curves y=1x2,y=ex.

The double integral is Dy2dA.

Calculation:

Use the graphing calculator to draw the given two curves and find the point of intersection of the curves as given below in the Figure 1.

From Figure 1, it is observed that the point of intersection is (0.71,0.4956) and (0,1). Also from the Figure 1. It is observed that x varies from 0.71 to 0. Thus, the value of the integral is,

0.711ex1x2y2dydx=0.710[y33]ex1x2dx=130.710[(1x2)3(ex)3]dx=130.710(13x2+3x4x6e3x)dx

Integrate with respect to x and apply the limit.

0.711ex1x2y2dydx=13[x3x33+3x55x77e3x3]0

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