   Chapter 15.2, Problem 13E

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.13. ∬ D x   d A , D is enclosed by the lines y = x, y = 0, x = 1

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)=x .

The region D is enclosed by the lines y = x, y = 0 and x = 1.

Definition used:

Region of type 1:

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 1 and y varies from 0 to x.

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

Df(x,y)dA=010xxdydx=01[xy]0xdx=01(x20)dx=01x2dx

Integrate with respect to x,

Df(x,y)dA=[x33]01=(130)=13

Thus, the value of the double integration when considering D as a region in type 1 and type 2 is 13

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started 