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Chapter 13: Q 69. (page 1068)

Let abe a constant. Prove that the equation of the planex=a isr=asec胃in cylindrical coordinates.

Short Answer

Expert verified

Use the transformation x=rcos胃,y=rsin胃,z=zto prove the mentioned relation.

Step by step solution

01

Given Information

Equation of plane in rectangular coordinates is given as x=a(ais constant).

02

Simplification

x=rcos胃,y=rsin胃,z=zare cylindrical coordinates.

Use x=rcos胃in the given equation to convert equation to cylindrical coordinates.

x=a

鈬�rcos胃=a

r=acos胃

r=asec胃

Hence proved.

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Most popular questions from this chapter

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.

Let f(x,y,z)be a continuous function of three variables, let localid="1650352548375" 惟yz={(y,z)|a鈮�y鈮�bandh1(y)鈮�z鈮�h2(y)}be a set of points in the yz-plane, and let localid="1650354983053" 惟={(x,y,z)|(y,z)鈭�惟yzandg1(y,z)鈮�x鈮�g2(y,z)}be a set of points in 3-space. Find an iterated triple integral equal to the triple integral localid="1650353288865" 鈭�惟f(x,y,z)dV. How would your answer change iflocalid="1650352747263" 惟yz={(y,z)|a鈮�z鈮�bandh1(z)鈮�x鈮�h2(z)}?

Evaluate each of the double integral in the exercise 37-54 as iterated integrals

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