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Chapter 13: Q. 39 (page 1004)

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}

Short Answer

Expert verified

The value of the integral is 113.33units

Step by step solution

01

Given information

We are given an integral

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

02

Use iterated integral

We get,

鈭�35鈭�-32(2-3x2+y2)dxdy=鈭�35[2x-x3+xy2]2-3dy=鈭�35(10-35+5y2)dy=[-25y+53y3]53=113.33units

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Most popular questions from this chapter

Let 惟be a lamina in the xy-plane. Suppose 惟is composed of n non-overlapping lamin忙 role="math" localid="1650321722341" 惟1,惟2,鈥�,惟n.Show that if the masses of these lamin忙 are m1,m2,鈥�,mnand the centers of masses are x炉1,y炉1,x炉2,y炉2,鈥�,x炉n,y炉n,then the center of mass of 惟is x炉,y炉,where

x炉=鈭�k=1nmkx炉k鈭�k=1nmkandy炉=鈭�k=1nmky炉k鈭�k=1nmk.

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)=xyWhereR={(x,y)|-2鈮�x鈮�3,-1鈮�y鈮�5}

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 at each point in Ris proportional to the distance of the point from the xy-plane.

(a) Without using calculus, explain why the x- and y-coordinates of the center of mass are x炉=2andy炉=32,respectively.

(b) Use an appropriate integral expression to find the z-coordinate of the center of mass.

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

Evaluate the sums in Exercises 23鈥�28.

鈭�j=1m鈭�k=1n(j-k)
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