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Chapter 13: Q 70. (page 1068)

Let $b$ be a constant. Prove that the equation of the plane $y = b$ is $r = b c s c 胃$ in cylindrical coordinates.

Short Answer

Expert verified

Conversion of equation is done using $x = r \cos 胃, y = r \sin 胃, z = z$ .

Step by step solution

Given Information

Equation of plane in rectangular coordinates is given by $y = b$ ( $b$ is constant)

Simplification

We know that

$x = r \cos 胃, y = r \sin 胃, z = z$

For the transformation, use $y = r \sin 胃$ in $y = b$

$r \sin 胃 = b$

$r = \frac{b}{\sin 胃}$

$鈬� r = b \csc 胃$

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Most popular questions from this chapter

Evaluate each of the double integrals in Exercises $37 - 54$ as iterated integrals.

localid="1650380493598" $鈭� {鈭�}_{R} \sin (2 x + y) d A,$

wherelocalid="1650380496793" $R = \{(x, y) | 0 鈮� x 鈮� \frac{蟺}{2} a n d 0 鈮� y 鈮� \frac{蟺}{2}\}$

$Express the sum \frac{2}{5} + \frac{2}{6} + \frac{2}{7} + \frac{2}{8} + \frac{4}{5} + \frac{4}{6} + \frac{4}{7} + \frac{4}{8} using double - summation notation .$

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

Show that if the masses and centers of masses of $惟_{1}$ and $惟_{2}$ are $m_{1}$ and $m_{2},$ and $({\overset{炉}{x}}_{1}, {\overset{炉}{y}}_{1}) and ({\overset{炉}{x}}_{2}, {\overset{炉}{y}}_{2})$ respectively, then the center of mass of $惟$ is $(\overset{炉}{x}, \overset{炉}{y}),$ where

$\overset{炉}{x} = \frac{m_{1} {\overset{炉}{x}}_{1} + m_{2} {\overset{炉}{x}}_{2}}{m_{1} + m_{2}} and \overset{炉}{y} = \frac{m_{1} {\overset{炉}{y}}_{1} + m_{2} {\overset{炉}{y}}_{2}}{m_{1} + m_{2}}$

In Exercises, let $S = \{(x, y) 鈭� x^{2} + y^{2} 鈮� 1 and x 鈮� 0\} .$

If the density at each point in S is proportional to the point鈥檚 distance from the origin, find the moments of inertia about the x-axis, the y-axis, and the origin. Use these answers to find the radii of gyration of S about the x-axis, the y-axis, and the origin.

In the following lamina, all angles are right angles and the density is constant:

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Let bbe a constant. Prove that the equation of the plane y=bis r=bcsc胃in cylindrical coordinates.

Short Answer

Step by step solution

Given Information

Simplification

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Let $b$ be a constant. Prove that the equation of the plane $y = b$ is $r = b c s c 胃$ in cylindrical coordinates.