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Chapter 13: Q.29 (page 991)

If the density at each point in Tis proportional to the point's distance from the x-axis, find the center of mass of T.

Short Answer

Expert verified

Area of the region bounded by the spiral and the x-axis is

A=蟺36

Step by step solution

01

given information

The objective of this problem is to find an iterated integral in polar coordinates that represents the area of the given region in the polar plane and then evaluate the integral.

The region is enclosed by the spiral r=胃and the x-axis on the interval localid="1650634059460" 0鈮�胃鈮�蟺

Plot of r=胃

Area of the region bounded by the spiral and the x- axis can be expressed as

A=鈭�0x鈭�0r-胃rdrd胃

Integrate with respect to rfirst.

A=鈭�0*r220ed胃A=鈭�0蟺胃2-02d胃A=胃360*

Area of the region bounded by the spiral and the x-axis is

A=蟺36

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Most popular questions from this chapter

Evaluate the triple integrals over the specified rectangular solid region.

鈭�R鈥�x2yzdV, whereR={(x,y,z)鈭�鈭�2鈮�x鈮�1,0鈮�y鈮�3,and1鈮�z鈮�5}

Evaluate the iterated integral :

鈭�01鈥�鈭�24鈥�鈭�鈭�15鈥�x+yz2dxdydz

In Exercises 61鈥�64, let R be the rectangular solid defined by

R=(x,y,z)|0鈮�a1鈮�x鈮�a2,0鈮�b1鈮�y鈮�b2,0鈮�c2鈮�z鈮�c2.

Assume that the density of R is uniform throughout.

(a) Without using calculus, explain why the center of

mass is a1+a22,b1+b22,c1+c22.

(b) Verify that a1+a22,b1+b22,c1+c22is the center of mass by using the appropriate integral expressions.

What is the difference between a triple integral and an iterated triple integral?

Explain how to construct a midpoint Riemann sum for a function of three variables over a rectangular solid for which each xi*,yj*,zk*is the midpoint of the subsolid role="math" localid="1650346869585" Rijk={x,y,k|xi-1鈮�xi*鈮�xi,yj-1鈮�yj*鈮�yjandzk-1鈮�zk*鈮�zk}. Refer either to your answer to Exercise 4or to Definition 13.14.
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