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Chapter 13: Q. 10 (page 991)

Identify the quantities determined by the integral expressions in Exercises 3-11. If x and y are both measured in centimeters and 蚁(x,y)is a density function in grams per square centimeter, give the units of the expression.

鈭�惟x2+y2蚁(x,y)dA

Short Answer

Expert verified

Moment of inertia about the origin will be measured in grams square centimeter.

Moment of inertia about the origin:

Ie=鈭�惟x2+y2蚁(x,y)dA

Step by step solution

01

Given information

The expression isIe=鈭�惟x2+y2蚁(x,y)dA

02

Simplification

The goal of this problem is to figure out what quantity the integral expression determines.

x and y are measured in centimeters and 蚁(x,y)is measured in grams per square centimeter. Give the units of expression.

The expression is:

鈭�惟x2+y2蚁(x,y)dA

The expression 鈭�惟x2+y2蚁(x,y)dArepresents the moment of inertia of the lamina with uniform density about the origin. Moment of inertia about the origin is also known as the polar moment.

If xand yare measured in centimeters and 蚁(x,y)is measured in grams per square centimeter. The moment of inertia around the origin will then be calculated in grams per square centimeter.

The required Moment of inertia about the origin:

Io=鈭�惟x2+y2蚁(x,y)dA

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Most popular questions from this chapter

Explain how to construct a Riemann sum for a function of two variables over a rectangular region.

Evaluate the triple integrals over the specified rectangular solid region.

鈭�R鈥�zsin鈦�xcos鈦�ydV,whereR=(x,y,z)鈭�0鈮�x鈮�蟺,3蟺2鈮�y鈮�2蟺,and1鈮�z鈮�3

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}.

Assuming that the density at each point in R is proportional to the distance of the point from the xy-plane, find the moment of inertia about the x-axis and the radius of gyration about the x-axis.

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.

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}
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