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Chapter 13: Q. 39 (page 1079)

Evaluate the double integrals in Exercises 39鈥�48. Use suitable transformations as necessary.

鈭�鈭�R(x-y)3dA, where R is the parallelogram with vertices (0, 0), (3, 0), (5, 2), and (2, 2).

Short Answer

Expert verified

鈭�鈭�R(x-y)3dA=812

Step by step solution

01

Draw the region

Plot the given points to form the region and name the vertices.

In the region R, the equations of boundary curves are,

AB:y=0BC:x-y=3CD:y=2DA:x-y=0

Consider the new set of variables defined as,

u=x-yv=y

Add the above equations,

u+v=x

Use these equations to determine the equation of each boundary of the region in terms of variables u and v.

AB:v=0BC:u=3CD:v=2DA:u=0

Plot these points on UV plane,

02

Find the integral

Set up the double integral for the region R' between u = 0 and u = 3.

鈭�鈭�R(x-y)3dA=鈭�03u3鈭�02dvdu=鈭�03u3v02du=2鈭�03u3du=2u4403=2脳814=812

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Most popular questions from this chapter

Howmanysummandsarein鈭�j=313鈭�k=520j2ejk?

Evaluate the sums in Exercises 23鈥�28.

鈭�j=13鈭�k=12jk

Evaluate each of the double integrals in Exercises 37-54as iterated integrals.

role="math" localid="1650327788023" 鈭�鈭�RysinxdA,

whererole="math" localid="1650327080219" R=x,y|0鈮�x鈮�蟺2and0鈮�y鈮�1

Identify the quantities determined by the integral expressions in Exercises 19鈥�24. If x, y, and z are all measured in centimeters and 蚁(x, y,z) is a density function in grams per cubic centimeter on the three-dimensional region , give the units of the expression.

鈭�惟鈥�x蚁(x,y,z)dV

LetR=x,y,z|a1猢�x猢�a2,b1猢�y猢�b2,c1猢�z猢�c2.Provethat:鈭�RdV=a2-a1b2-b1c2-c1.WhatistherelationshipbetweenRandtheproducta2-a1b2-b1c2-c1.
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