Q 75. Let R聽be the radius of the base... [FREE SOLUTION]

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Chapter 13: Q 75. (page 1068)

Let $R$ be the radius of the base of a cone and $h$ be the height of the cone. Use spherical coordinates to set up and evaluate a triple integral proving that the volume of the cone is $\frac{1}{3} 蟺 R^{2} h$ .

Short Answer

Expert verified

This is done by finding the volume integral and substituting the limits of $(蚁, 蠒, 胃)$ .

Step by step solution

Given Information

It is given that $R$ is radius of cone and $h$ is height of cone.

Evaluation of limits

The limits of spherical coordinates are described as

$0 鈮� 蠒鈮� \tan^{- 1} (\frac{R}{b}), 0 鈮� 胃鈮� 2 蟺, 0 鈮� 蚁鈮� h \sec 蠒$ .

Evaluating the Volume

Using spherical coordinates, the volume is evaluated as

$V = {鈭�}_{0}^{{Tan}^{- 1} (\frac{R}{h})} {鈭�}_{0}^{2 蟺} {鈭�}_{0}^{h \sec 蠒} 蚁^{2} \sin 胃 d 蚁 d 胃 d 蠒$

$= {鈭�}_{0}^{{Tan}^{- 1} (\frac{R}{h})} {鈭�}_{0}^{2 蟺} ({鈭�}_{0}^{h \sec 蠒} 蚁^{2} d 蚁) \sin 胃 d 胃 d 蠒$

$= {鈭�}_{0}^{{Tan}^{- 1} {(\frac{R}{h})}^{2 蟺}} {鈭�}_{0} {[\frac{蚁^{3}}{3}]}_{0}^{h \sec 蠒} \sin 蠁 d 胃 d 蠒$

$= {鈭�}_{0}^{{Tan}^{- 1} {(\frac{R}{h})}^{2 蟺}} {鈭�}_{0}^{[} [\frac{(h \sec 蠒)^{3}}{3} - 0] \sin 蠁 d 胃 d 蠒$

$= \frac{h^{3}}{3} {鈭�}_{0}^{{Tan}^{- 1} (\frac{R}{h})} ({鈭�}_{0}^{2 蟺} d 胃) \frac{1}{\cos^{3} 蠒} \sin 蠒 d 蠒$

$= \frac{h^{3}}{3} {鈭�}_{0}^{{Tan}^{- 1} (\frac{R}{h})} (胃)_{0}^{2 蟺} \frac{\sin 蠒}{\cos^{3} 蠒} d 蠒$

$= \frac{h^{3}}{3} {鈭�}_{0}^{{Tan}^{- 1} (\frac{R}{n})} (2 蟺) \tan 蠒 \sec^{2} 蠒 d 蠒$

$= \frac{h^{3}}{3} (2 蟺) {鈭�}_{0}^{{Tan}^{- 1} (\frac{R}{h})} \tan 蠒 \sec^{2} 蠒 d 蠒$

$= \frac{h^{3}}{3} (2 蟺) {[\frac{(\tan 蠒)^{2}}{2}]}_{0}^{{Tan}^{- 1} (\frac{R}{h})}$

Using limits, we get

$V = \frac{h^{3}}{3} (2 蟺) [\frac{{({tantan}^{- 1} (\frac{R}{h}))}^{2}]}{2}]$

$V = \frac{h^{3}}{3} (2 蟺) [\frac{{(\frac{R}{h})}^{2}}{2}]$

$V = \frac{1}{3} 蟺 R^{2} h$

Hence, proved.

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91影视

Let $R$ be the radius of the base of a cone and $h$ be the height of the cone. Use spherical coordinates to set up and evaluate a triple integral proving that the volume of the cone is $\frac{1}{3} 蟺 R^{2} h$ .

Short Answer

Step by step solution

Given Information

Evaluation of limits

Evaluating the Volume

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91影视

Let Rbe the radius of the base of a cone and hbe the height of the cone. Use spherical coordinates to set up and evaluate a triple integral proving that the volume of the cone is 13蟺R2h.

Short Answer

Step by step solution

Given Information

Evaluation of limits

Evaluating the Volume

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Geometry

Logic and Functions

Applied Mathematics

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Statistics

Study anywhere. Anytime. Across all devices.

Let $R$ be the radius of the base of a cone and $h$ be the height of the cone. Use spherical coordinates to set up and evaluate a triple integral proving that the volume of the cone is $\frac{1}{3} 蟺 R^{2} h$ .