Chapter 13: Q. 36 (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.
$\int_{0}^{π / 4} \int_{0}^{2 π} \int_{0}^{4 s e c θ} p^{2} \sin ϕ d p d θ d ϕ$

Short Answer

Expert verified

$\int_{0}^{π / 4} \int_{0}^{2 π} \int_{0}^{4 s e c θ} p^{2} \sin ϕ d p d θ d ϕ = 0$

Step by step solution

Given

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

\int_{0}^{π / 4} \int_{0}^{2 π} \int_{0}^{4 s e c θ} p^{2} \sin ϕ d p d θ d ϕ

Calculate the volume

$V = \int_{0}^{π / 4} \int_{0}^{2 π} \int_{0}^{4 s e c θ} p^{2} \sin ϕ d p d θ d ϕ V = \int_{0}^{π / 4} \sin ϕ d ϕ \int_{0}^{2 π} {[\frac{p^{3}}{3}]}_{0}^{4 s e c θ} d θ V = \frac{64}{3} \int_{0}^{π / 4} \sin ϕ d ϕ [\int_{0}^{2 π} s e c^{3} θ d θ] V = \frac{64}{3} \int_{0}^{π / 4} \sin ϕ d ϕ {[s e c θ \cdot \tan θ + \log |s e c θ + \tan θ|]}_{0}^{2 π} V = \frac{64}{3} \int_{0}^{π / 4} \sin ϕ d ϕ [(s e c 2 π \cdot \tan 2 π + \log |s e c 2 π + \tan 2 π|) - (s e c 0 \cdot \tan 0 + \log |s e c 0 + \tan 0|)] V = \frac{64}{3} \int_{0}^{π / 4} \sin ϕ d ϕ [0 - 0] V = 0 \times \frac{64}{3} \int_{0}^{π / 4} \sin ϕ V = 0$

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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.
$\int_{0}^{π / 4} \int_{0}^{2 π} \int_{0}^{4 s e c θ} p^{2} \sin ϕ d p d θ d ϕ$

Short Answer

Step by step solution

Given

Calculate the volume

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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/4∫02π∫04secθp2sinϕdpdθdϕ

Short Answer

Step by step solution

Given

Calculate the volume

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Most popular questions from this chapter

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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.
$\int_{0}^{π / 4} \int_{0}^{2 π} \int_{0}^{4 s e c θ} p^{2} \sin ϕ d p d θ d ϕ$