   Chapter 15.2, Problem 14E

Chapter
Section
Textbook Problem

Express D as a region of type I and also as a region of type II. Then evaluate the double integral in two ways.14. ∬ D x y   d A , D is enclosed by the curves y = x2, y = 3x

To determine

To express: The given region D as of type 1 and type 2 and then evaluate the double integral.

Explanation

Given:

The function is f(x,y)=xy .

The region D is enclosed by the curves y=x2,y=3x .

Definition used:

Region of type 1:

A plane region D is said to be of type 1 if it lies between two continuous functions of x.

That is, D={(x,y)|axb,g1(x)yg2(x)} , where g1(x) and g2(x) are the continuous functions of x.

Region of type 2:

A plane region D is said to be of type 2 if it lies between two continuous functions of y.

That is, D={(x,y)|ayb,h1(y)xh2(y)} , where h1(y) and h2(y) are the continuous functions of y.

If the region D is of type 1, then the region that has to be evaluated is given below in the Figure 1.

Therefore, from Figure 1, it is observed that x varies from 0 to 3and y varies from x2 to 3x.

So, the value of double integral is computed as follows.

Df(x,y)dA=03x23xxydydx=03[xy22]x23xdx=03(x(3x)22x(x2)22)dx=03(9x32x52)dx

Integrate with respect to x,

Df(x,y)dA=[9x42(4)x6

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