   Chapter 15.2, Problem 34E

Chapter
Section
Textbook Problem

Find the approximate volume of the solid in the first octant that is bounded by the planes y = x, z = 0, and z = x and the cylinder y = cos x. (Use a graphing device to estimate the points of intersection.)

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=x,y=cosx,z=0 .

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 approximately 0.739

The value of the given double integral is,

V=RzdA=00.739xcosxxdydx

Integrate with respect to y and apply the limit.

V=00.739[xy]xcosxdx=00.739[(xcosxx(x))]dx=00.739[xcosxx2]dx=00.739xcosxdx00

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