   Chapter 15.1, Problem 44E

Chapter
Section
Textbook Problem

Graph the solid that lies between the surface z = 2xy/(x2 + 1) and the plane z = x + 2y and is bounded by the planes x = 0, x = 2, y = 0, and y = 4. Then find its volume.

To determine

To sketch: The graph of the given solid that lies between the given surface and the plane and is bounded by the planar region; also find the volume of the solid.

Explanation

Formula used:

If g(x) is the function of x and h(y) is the function of y then,

abcdg(x)h(y)dydx=abg(x)dxcdh(y)dy (1)

The volume of the solid, V=RzdA .

Here, z is the given function.

Given:

The surface is z=2xy1+x2 .

The planes are z=x+2y,x=0,x=2&y=0,y=4 .

Calculation:

Let G1 denote the plane z=x+2y and G2 denote the surface z=2xy1+x2

Then the given solid lies below the surface G2 and lies above the plane G1 .

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

The rectangular region is R=[0,2]×[0,4] and the solid lies below z=x+2y .

Obtain the volume of the solid by using the equation (1).

V=RzdAR(2xy1+x2)dA=0204(x+2y)dydx0204(2xy1+x2)

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