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Chapter 13: Q 53. (page 1067)

Find the volume using integrals:

The region in the next figure which is bounded below by the xy-plane, bounded above by the hyperboloid with equation x2+y2-z2=1and inside the cylinder with equationx2+y2=5.

Short Answer

Expert verified

The required Volume isV=16蟺3units.

Step by step solution

01

Given Information

The given equations are x2+y2-z2=1andx2+y2=5.

02

Evaluation of limits

The relation between rectangular and spherical coordinates are as below:

r=x2+y2,tan胃=yx,z=z

And

x=rcos胃,y=rsin胃,z=z

Rectangular coordinates are x2+y2-z2=1and x2+y2=5

Cylindrical coordinates are r2-z2=1and r2=5

Limits for Cartesian coordinates are:

x2+y2-z2=1鈬�z=r2-1(Equation for xyplane is z=0)

x2+y2=5鈬�r=5and z=0鈬�r=1

Limits for Cylindrical coordinates are:

0鈮�z鈮�r2-1,1鈮�r鈮�5,0鈮�胃鈮�2蟺

03

Calculation of Volume

Required Volume is given by

V=鈭�胃=02蟺鈭�r=15鈭�z=0r2-1rdzdrd胃

V=鈭�胃=02蟺12鈭�r=152rr2-1drd胃

V=鈭�胃=02蟺12r2-13/23/2r=1r=5d胃

V=13鈭�胃=02蟺(8-0)d胃

V=83(胃)02蟺

V=832蟺

Hence, V=16蟺3units

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Most popular questions from this chapter

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

鈭�鈭�R(2-3x2+y3)dAWhereR={(x,y)|-3鈮�x鈮�2,3鈮�y鈮�5}

Evaluate each of the double integrals in Exercises37-54as iterated integrals.

localid="1650380493598" 鈭�鈭�Rsin2x+ydA,

wherelocalid="1650380496793" R=x,y|0鈮�x鈮�蟺2and0鈮�y鈮�蟺2

Find the masses of the solids described in Exercises 53鈥�56.

The solid bounded above by the paraboloid with equation z=8-x2-y2and bounded below by the rectangle R={(x,y,0)|1鈮�x鈮�2and0鈮�y鈮�2}in the xy-plane if the density at each point is proportional to the square of the distance of the point from the origin.

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�03鈭�01-y/3鈭�02-(2/3)y-2zf(x,y,z)dxdzdy
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