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Chapter 13: Q. 60 (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)=-2x2y3WhereR={(x,y)|-2鈮�x鈮�3,-1鈮�y鈮�5}

Short Answer

Expert verified

The value of the integral is1686.66cubicunits

Step by step solution

01

Given information

We are given a rectangular region

f(x,y)=-2x2y3WhereR={(x,y)|-2鈮�x鈮�3,-1鈮�y鈮�5}

02

Find the volume

We have,

f(x,y)>0whenR1={(x,y)|-2鈮�x鈮�0,-1鈮�y鈮�0}Andf(x,y)<0whenR2={(x,y)|0鈮�x鈮�3,0鈮�y鈮�5}

Also we have,

鈭�R-2x2y3dA=鈭�R1-2x2y3dA-鈭�R2-2x2y3dA=鈭�-10鈭�-20-2x2y3dxdy-鈭�03鈭�05-2x2y3dxdy=[-2x3y412]-[-2x3y412]=-1612+[2025012]=1686.66cubicunits

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Most popular questions from this chapter

Expressthesum25+26+27+28+45+46+47+48usingdouble-summationnotation.

What is the difference between a double integral and an iterated integral?

In Exercises 61鈥�64, let R be the rectangular solid defined by R=(x,y,z)|0鈮�a1鈮�x鈮�a2,0鈮�b1鈮�y鈮�b2,0鈮�c2鈮�z鈮�c2.

Assume that the density of R is uniform throughout, and find the moment of inertia about the x-axis and the radius of gyration about the x-axis.

Evaluate the triple integrals over the specified rectangular solid region.

鈭�R鈥�(x+2y+3z)dV, whereR={(x,y,z)鈭�0鈮�x鈮�4,1鈮�y鈮�5,and2鈮�z鈮�7}

Discuss the similarities and differences between the definition of the definite integral found in Chapter 4 and the definition of the double integral found in this section.
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