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Chapter 13: Q 28. (page 1066)

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

Short Answer

Expert verified

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

$(r, 胃, z) = (3 \sqrt{2}, \frac{蟺}{4}, 3 \sqrt{2})$

Step by step solution

01

Given Information

The given expression is $(蚁, 胃, 蠒) = (6, \frac{蟺}{4}, \frac{蟺}{4})$

02

To find rectangular coordinates

As $蚁 = 6, 胃 = \frac{蟺}{4}, 蠒 = \frac{蟺}{4}$

We know that

$x = 蚁 \sin 蠒 \cos 胃, y = 蚁 \sin 蠒 \sin 胃, z = 蚁 \cos 蠒$

Substituting values, we get

$x = 6 \sin \frac{蟺}{4} \cos \frac{蟺}{4}, y = 6 \sin \frac{蟺}{4} \sin \frac{蟺}{4}, z = 6 \cos \frac{蟺}{4}$

$鈬� x = 6 \frac{1}{\sqrt{2}} \frac{1}{\sqrt{2}}, y = 6 \frac{1}{\sqrt{2}} \frac{1}{\sqrt{2}}, z = 6 \frac{1}{\sqrt{2}}$

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

Hence, rectangular coordinates are $(x, y, z) = (3, 3, 3 \sqrt{2})$

03

To find cylindrical coordinates

To find cylindrical coordinates from spherical coordinates, we use

$r = 蚁 \sin 蠒, 胃 = 胃, z = 蚁 \cos 蠒$

Using given values of spherical coordinates, we get

$r = 6 \sin \frac{蟺}{4}, 胃 = \frac{蟺}{4}, z = 6 \cos \frac{蟺}{4}$

$r = 6 \frac{1}{\sqrt{2}}, 胃 = \frac{蟺}{4}, z = 6 \frac{1}{\sqrt{2}}$

$鈬� r = 3 \sqrt{2}, 胃 = \frac{蟺}{4}, z = 3 \sqrt{2}$

Hence, cylindrical coordinates are $(r, 胃, z) = (3 \sqrt{2}, \frac{蟺}{4}, 3 \sqrt{2})$

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Most popular questions from this chapter

Discuss the similarities and differences between the definition of the definite integral found in Chapter 4 and the definition of the double integral found in this section.

Explain why it would be difficult to evaluate the double integrals in Exercises 18 and 19 as iterated integrals.

$鈭� {鈭�}_{R} \cos (x y) d A w h e r e R = \{(x, y) I \frac{蟺}{4} 鈮� x 鈮� \frac{蟺}{2} a n d \frac{蟺}{2} 鈮� y 鈮� 蟺\}$

Let $f (x)$ be an integrable function on the rectangular solid $R = \{(x, y, z) | a_{1} 猢� x 猢� a_{2}, b_{1} 猢� y 猢� b_{2}, c_{1} 猢� z 猢� c_{2}\}$ , and let $魏鈭� 鈩� .$ Use the definition of the triple integral to prove that:

${鈭�}_{R} 魏 f (x, y, z) d V = 魏 {鈭�}_{R} f (x, y, z) d V .$

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

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

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