Chapter 13: Q. 34 (page 1083)

Evaluating triple integrals: Each of the triple integrals that follows represents the volume of a solid. Sketch the solid and evaluate the integral.
$\int_{0}^{π} \int_{0}^{2} \int_{0}^{r \sin θ} r d z d r d θ$

Short Answer

Expert verified

$\int_{0}^{π} \int_{0}^{2} \int_{0}^{r \sin θ} r d z d r d θ = \frac{16}{3}$

Step by step solution

Given

We have to find the volume by using the triple integral.

\int_{0}^{π} \int_{0}^{2} \int_{0}^{r \sin θ} r d z d r d θ

Evaluate the integral

$V = \int_{0}^{π} \int_{0}^{2} [\int_{0}^{r \sin θ} r d z] d r d θ V = \int_{0}^{π} \int_{0}^{2} [r (r \sin θ - 0)] d r d θ V = \int_{0}^{π} \int_{0}^{2} r^{2} \sin θ d r d θ V = \int_{0}^{π} \sin θ d θ \int_{0}^{2} r^{2} d r V = \int_{0}^{π} \sin θ d θ {[\frac{r^{3}}{3}]}_{0}^{2} V = \frac{8}{3} \int_{0}^{π} \sin θ d θ V = \frac{8}{3} {[- \cos θ]}_{0}^{π} V = \frac{8}{3} [- \cos π + \cos 0] V = \frac{8}{3} [1 + 1] V = \frac{16}{3} cubic units$

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Evaluating triple integrals: Each of the triple integrals that follows represents the volume of a solid. Sketch the solid and evaluate the integral.
$\int_{0}^{π} \int_{0}^{2} \int_{0}^{r \sin θ} r d z d r d θ$

Short Answer

Step by step solution

Given

Evaluate the integral

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Evaluating triple integrals: Each of the triple integrals that follows represents the volume of a solid. Sketch the solid and evaluate the integral. ∫0π∫02∫0rsinθrdzdrdθ

Short Answer

Step by step solution

Given

Evaluate the integral

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Statistics

Discrete Mathematics

Applied Mathematics

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Geometry

Study anywhere. Anytime. Across all devices.

Evaluating triple integrals: Each of the triple integrals that follows represents the volume of a solid. Sketch the solid and evaluate the integral.
$\int_{0}^{π} \int_{0}^{2} \int_{0}^{r \sin θ} r d z d r d θ$