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Chapter 13: Q 26. (page 1066)

Give the rectangular and spherical coordinates for the point with cylindrical coordinates $(4, \frac{蟺}{3}, 6)$

Short Answer

Expert verified

$(x, y, z) = (2, 2 \sqrt{3}, 6)$

$(蚁, 胃, 蠒) = ({\sqrt{52}, \frac{蟺}{3}, \tan}^{- 1} \frac{2}{63})$

Step by step solution

01

Given Information

The given expression is $(r, 胃, z) = (4, \frac{蟺}{3}, 6)$

02

To find rectangular coordinates

As $r = 4, 胃 = \frac{蟺}{3}, z = 6$

We know that for converting cylindrical to rectangular coordinates, we use

$x = r \cos 胃, y = r \sin 胃, z = z$

localid="1652288468915" $x = 4 \cos \frac{蟺}{3}, y = 4 \sin \frac{蟺}{3}, z = 6$

$x = 4 \frac{1}{2}, y = 4 \frac{\sqrt{3}}{2}, z = 6$

$x = 2, y = 2 \sqrt{3}, z = 6$

Hence, $(x, y, z) = (2, 2 \sqrt{3}, 6)$

03

To find spherical coordinates

For converting cylindrical to spherical coordinates, we use

$蚁 = {\sqrt{r^{2} + z^{2}}, 胃 = 胃, 蠒 = \tan}^{- 1} \frac{r}{z}$

Using given values,

$蚁 = {\sqrt{4^{2} + 6^{2}}, 胃 = \frac{蟺}{3}, 蠒 = \tan}^{- 1} \frac{4}{6}$

$蚁 = {\sqrt{52}, 胃 = \frac{蟺}{3}, 蠒 = \tan}^{- 1} \frac{2}{3}$

$鈬� 蚁 = {2 \sqrt{13}, 胃 = \frac{蟺}{3}, 蠒 = \tan}^{- 1} [\frac{2}{3}]$

Hence, $(蚁, 胃, 蠒) = (2 \sqrt{13}, {\frac{蟺}{3}, \tan}^{- 1} [\frac{2}{3}])$

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Most popular questions from this chapter

Evaluate the iterated integral :

${鈭�}_{1}^{3} 鈥� {鈭�}_{鈭� 1}^{5} 鈥� {鈭�}_{0}^{3} 鈥� x y^{2} z d z d y d x$

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

$R = \{(x, y, z) | 0 鈮� a_{1} 鈮� x 鈮� a_{2}, 0 鈮� b_{1} 鈮� y 鈮� b_{2}, 0 鈮� c_{2} 鈮� z 鈮� c_{2}\} .$

Assume that the density of R is uniform throughout.

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

mass is $(\frac{a_{1} + a_{2}}{2}, \frac{b_{1} + b_{2}}{2}, \frac{c_{1} + c_{2}}{2}) .$

(b) Verify that $(\frac{a_{1} + a_{2}}{2}, \frac{b_{1} + b_{2}}{2}, \frac{c_{1} + c_{2}}{2})$ is the center of mass by using the appropriate integral expressions.

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

Evaluate the triple integrals over the specified rectangular solid region.

$\begin{aligned} {鈭�}_{R} 鈥� \ln 鈦� (x y z^{2}) d V, where \\ R = {(x, y, z) 鈭� 1 鈮� x 鈮� 3, 1 鈮� y 鈮� e, and 1 鈮� z 鈮� 2} \end{aligned}$

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

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