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Chapter 5: Q22P (page 247)

Above the triangle with vertices (0,2),(1,1) and (2,2) , and under the surface z = xy.

Short Answer

Expert verified

The required solution is 53

Step by step solution

01

Definition of double integral

The double integral of f(x,y)over the area Ain the (x,y)plane as the limit of this sum, and we write it as鈭�鈭�Af(x,y)dxdy

02

Calculation of the volume under the curve

The volume above the triangle with vertices (0,2), (1,1) and (2,2), and under the surface z = xy .

First, integrate over z :

|=鈭�x=12鈭�y=2-xx鈭�z=0xydzdxdy=鈭�12dx鈭�2-xxdyxy

03

Further calculation of the volume of the bounded region

Now, integrate over y:

|=鈭�12xdx12y22-xx=12鈭�12x(x2-4+4x-x2)dx

04

Further calculation of the volume of the bounded region

Now, integrate over x:

|=2x33-12x212=53

Therefore, the value is52

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Most popular questions from this chapter

In the problems of this section, set up and evaluate the integrals by hand and check your results by computer鈭�x=12鈭�y=218xydxdy.

The volume inside a sphere of radius risV=43蟿蟿谤3. ThendV=4蟿蟿谤2dr=Adrwhereis the area of the sphere. What is the geometrical meaning of the fact that the derivative of the volume is the area? Could you use this fact to find the volume formula given the area formula?

Write a triple integral in cylindrical coordinates for the volume inside the cylinder x2+y2=4and between z=2x2+y2 and the (x,y) plane. Evaluate the integral.

鈭�x=12鈭�y=x2x鈭�z=0y-xdzdydx

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