   Chapter 15.1, Problem 37E

Chapter
Section
Textbook Problem

Find the volume of the solid that lies under the plane 4x + 6y − 2z + 15 = 0 and above the rectangle R = {{x, y}| −1 ≤ x ≤ 2, −1 ≤ y ≤ 1}.

To determine

To find: The volume of the solid that lies under the plane and above the rectangular region.

Explanation

Calculation:

Formula used:

The volume of the solid, V=RzdA , where, z is the given function.

Given:

The plane is 4x+6y2z+15=0 .

The rectangular region is, R={(x,y)|1x2,1y1} .

Calculation:

Express the given plane equation as follows:

2z=4x6y15z=(42)x(62)y(152)z=2x+3y+152

The volume of the solid is computed as follows.

V=RzdA=1211(2x+3y+152)dydx

First, compute the integral with respect to y

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