   Chapter 15.2, Problem 33E

Chapter
Section
Textbook Problem

Use a graphing calculator or computer to estimate the x-coordinates of the points of intersection of the curves y = x4 and y = 3x − x2. If D is the region bounded by these curves, estimate ∬ D x   d A .

To determine

To estimate: The x-coordinate of the point of intersection of the curves and evaluate the double integral.

Explanation

Given:

The surface is z=x .

The curves are y=x4,y=3xx2 .

Formula used:

The value of the given double integral is, V=RzdA , where z is the given function.

Calculation:

Use online graphing calculator and sketch the required graph as shown below in Figure 1.

The x-coordinates of the point of intersection of two curves is at 0 and approximately 1.213.

The value of the given double integral is,

V=RzdA=01.213x43xx2xdydx

Integrate with respect to y and apply the limit.

V==01.213[xy]x43xx2dx=01.213[(x(3xx2)x(x4))]dx=01

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