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Chapter 13: Q. 1 (page 1082)

Using the definition to evaluate a double integral: Evaluate the given double integrals as a limit of a Riemann sum. For each integral, let R=(x,y)|0鈮x鈮2and1鈮y鈮4.

鈭(x+2y)dA

Short Answer

Expert verified

鈭(x+2y)dA=36squareunits

Step by step solution

01

Definition of double integral

Let R=(x,y)|a鈮x鈮bandc鈮y鈮dand f(x,y)is a continuous function on R, Then

鈭f(x,y)dA=鈭ab鈭cdf(x,y)dxdy

02

Evaluate the integral

鈭(x+2y)dA=鈭14鈭02(x+2y)dxdy=鈭14鈭02(x+2y)dxdy=鈭14dyx22+2xy02=鈭14dy222+2脳2脳y=鈭14(2+4y)dy=2y+4y2214=8+32-2+2=40-4=36

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Most popular questions from this chapter

Evaluate the triple integrals over the specified rectangular solid region.

鈭R鈥ln鈦xyz2dV, whereR={(x,y,z)鈭1鈮x鈮3,1鈮y鈮e,and1鈮z鈮2}

Howmanysummandsarein鈭j=313鈭k=520j2ejk?

Let f(x)be an integrable function on the rectangular solid R=x,y,z|a1猢x猢a2,b1猢y猢b2,c1猢z猢c2, and let 魏鈭鈩.Use the definition of the triple integral to prove that:

鈭R魏f(x,y,z)dV=魏鈭Rf(x,y,z)dV.

Evaluate each of the double integral in the exercise 37-54 as iterated integrals

鈭鈭R(3-x+4y)dAwhereR={(x,y)|0鈮x鈮1,-1鈮y鈮3}

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z蚁(x,y,z)dV
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