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Chapter 13: Q. 44 (page 1015)

Find the volumes of the solids described in Exercises 41鈥44. The solid bounded above by the hyperboloid with equationz=x2鈭y2and bounded below by the square with vertices:(2,2,鈭4),(2,鈭2,鈭4),(鈭2,鈭2,鈭4),(鈭2,2,鈭4).

Short Answer

Expert verified

The volume of the described solid is: 0

Step by step solution

01

Step 1. Given Information  

The solid bounded above by the hyperboloid, z=x2鈭y2

The solid bounded below by the square with vertices:

(2,2,鈭4),(2,鈭2,鈭4),(鈭2,鈭2,鈭4),and(鈭2,2,鈭4).

The region is:R={(x,y)|-2鈮x鈮2and-2鈮y鈮2}

02

Step 2. Finding the volume of given solid

The double integral to find the volume of the given solid is given by,

鈭-22鈭-22x2-y2dydx=鈭-22x2y-y33-22dx=鈭-22x2(2-(-2)-23-(-2)33dx=鈭-224x2-163dx=4x33-16x3-22=4(23-(-2)3)3-16(2-(-2))3=643-643=0

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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Rsin(x+2y)dAWhereR={(x,y)|0鈮x鈮蟺,0鈮y鈮蟺2}

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}

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

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x2+z2蚁(x,y,z)dV

Evaluate the sums in Exercises 23鈥28.

鈭j=13鈭k=12jk
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