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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∫∫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 :

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

03

Further calculation of the volume of the bounded region

Now, integrate over y:

|=∫12xdx12y22-xx=12∫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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∫y=01∫x=y21exxdxdy

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