   Chapter 15.2, Problem 62E

Chapter
Section
Textbook Problem

Find the averge value of f over the region D.62. f(x, y) = x sin y, D is enclosed by the curves y = 0, y = x2, and x = 1

To determine

To find: The average value of given function over the region D.

Explanation

Formula used:

The average value of given function over the region D is,

fave=1A(D)Df(x,y)dA , where A(D) is the area of the given region D.

Given:

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

The region D is enclosed by the curves y=0,y=x2,x=1 .

Calculation:

The area of the region D is,

A(D)=01x2dx=[x33]01=13 .

The average value of given function is,

fave=113010x2xsinydydx=301[xcosy]0x2dx

Apply the limit value of y,

fave=301[xcos(x2)xcos(0)]dx=301xcosx2dx+301xdx

Integrate with respect to x

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