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Chapter 13: Q. 57 (page 1005)

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

f(x,y)=y3exy2WhereR={(x,y)|0鈮�x鈮�2,-2鈮�y鈮�3}

Short Answer

Expert verified

The volume is16414245cubicunits

Step by step solution

01

Given information

We are given an integral as

f(x,y)=y3exy2WhereR={(x,y)|0鈮�x鈮�2,-2鈮�y鈮�3}

02

Evaluate the integral

We have

f(x,y)>0whenR1={(x,y)|0鈮�x鈮�2,0鈮�y鈮�3}andf(x,y)<0whenR2={(x,y)|0鈮�x鈮�2,-2鈮�y鈮�0}

Therefore

鈭�Ry3exy2dA=鈭�R1y3exy2dA-鈭�R2y3exy2dA=鈭�03鈭�02y3exy2dxdy-鈭�-20鈭�02y3exy2dxdy=鈭�03[yexy2]20dy-鈭�-20[yexy2]20dy=鈭�03(ye2y2-y)dy-鈭�-20(ye2y2-y)dy=16414245cubicunits

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Most popular questions from this chapter

Describe the three-dimensional region expressed in each iterated integral in Exercises 35鈥�44.

鈭�-33鈭�-9-x29-x2鈭�-33f(x,y,z)dzdydx

Evaluate the sums in Exercises 23鈥�28.

鈭�j=13鈭�k=12jk

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

localid="1649936867482" 鈭�xx2ydA

whereR={(x,y)鈭�1鈮�x鈮�3and0鈮�y鈮�2}

Explain why using an iterated integral to evaluate a double integral is often easier than using the definition of the double integral to evaluate the integral.

Describe the three-dimensional region expressed in each iterated integral in Exercises 35鈥�44.

鈭�02鈭�03鈭�4x/34f(x,y,z)dzdxdy
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