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Chapter 5: Q19P (page 247)

Above the square with vertices at, (0,0), (2,0),(0,2) and (2,2) and under the plane z = 8-x+y.

Short Answer

Expert verified

The required solution is 32.

Step by step solution

01

Definition of double integral

The double integral of f(x,y)over the areaA in the (x,y)plane as the limit of this sum, and we write it as∫∫Af(x,y)dxdy.

02

Calculation of the volume under the curve

The volume above the square with vertices at (0,0),(2,0),(0,2) and (2,2), and under the plane z=8-x+y.

First, integrate over z:

|=∫x=02∫y=02∫z=08-x+ydzdxdy=∫02dy∫02dx8-x+y

03

Further calculation of the volume of the bounded region

Now, integrate over x:

|=∫02dy8x-12x2+yx02=∫02dy(16-2+2y)

04

Further calculation of the volume of the bounded region

Now, integrate over y:

|=∫02dy(16-2+2y)=(14y+y2)02=32

Therefore, the value is 32

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