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Chapter 13: Q. 62 (page 1005)

Find the volume between the graph of the given function and the xy-plane over the specified rectangle in the xy-plane

f(x,y)=yxeyWjereR={(x,y)|1鈮�x鈮�e,0鈮�y鈮�2}

Short Answer

Expert verified

Volume of the integral ise2-1cubicunits

Step by step solution

01

Given information

We are given a region

f(x,y)=yxeyWjereR={(x,y)|1鈮�x鈮�e,0鈮�y鈮�2}

02

Evaluate the integral

We have

鈭�RyxeydA=鈭�02鈭�1eyxeydxdy=鈭�02[lnx]e1路yeydy=鈭�02yeydy=[yey-ey]20 =e2-1

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Most popular questions from this chapter

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

Use Definition 13.4to evaluate the double integrals in Exercises 29鈥�32.

鈭�RxydA

where

R={(x,y)鈭�0鈮�x鈮�2&1鈮�y鈮�4}

In Exercises 45鈥�52, rewrite the indicated integral with the specified order of integration.

Exercise 41 with the order dy dx dz.

Expressthesum3e4+3e9+3e16+4e4+4e9+4e16usingdouble-summationnotation.

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.
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