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Chapter 13: Q.26 (page 1082)

Evaluating triple integrals: Each of the triple integrals that follow represents the volume of a solid. Sketch the solid and evaluate the integral.

鈭�15鈭�-20鈭�-13dzdydx

Short Answer

Expert verified

The obtained integral is 32.

Step by step solution

01

Step 1. GIven information.

Consider the given information.

鈭�15鈭�-20鈭�-13dzdydx

02

Step 2. Graph the solid.

By using the limits, sketch the given solid.

-1鈮�z鈮�3-2鈮�y鈮�01鈮�x鈮�5

And,

03

Step 3. Calculate the integral.

Integrate by using limit and solve.

鈭�15鈭�-20鈭�-13dzdydx=鈭�15鈭�-20z-13dydx=鈭�15鈭�-203--1-13dydx=4鈭�15y-20dx=4鈭�150--2dx=8x15=85-1=32

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Most popular questions from this chapter

Evaluate the iterated integral :

鈭�01鈥�鈭�24鈥�鈭�鈭�15鈥�x+yz2dxdydz

Evaluate the sums in Exercises 23鈥�28.
鈭�j=13鈭�k=14(3j-4k)

Let 惟be a lamina in the xy-plane. Suppose 惟is composed of two non-overlapping lamin 惟1and 惟2, as follows:

Show that if the masses and centers of masses of 惟1and 惟2are m1and m2,and x炉1,y炉1andx炉2,y炉2respectively, then the center of mass of 惟is x炉,y炉,where

x炉=m1x炉1+m2x炉2m1+m2andy炉=m1y炉1+m2y炉2m1+m2

In Exercises 57鈥�60, let R be the rectangular solid defined by

R = {(x, y, z) | 0 鈮� x 鈮� 4, 0 鈮� y 鈮� 3, 0 鈮� z 鈮� 2}.

Assume that the density ofR is uniform throughout.

(a) Without using calculus, explain why the center of mass is (2, 3/2, 1).

(b) Verify that the center of mass is (2, 3/2, 1), using the appropriate integral expressions.

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}
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