Chapter 13: Q. 35 (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.
$\int_{0}^{π} \int_{0}^{2 π} \int_{0}^{5} p^{2} \sin ϕ d p d θ d ϕ$

Short Answer

Expert verified

$\int_{0}^{π} \int_{0}^{2 π} \int_{0}^{5} p^{2} \sin ϕ d p d θ d ϕ = \frac{500}{3} π$

Step by step solution

Given

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

\int_{0}^{π} \int_{0}^{2 π} \int_{0}^{5} p^{2} \sin ϕ d p d θ d ϕ

Evaluate integral

$V = \int_{0}^{π} \int_{0}^{2 π} \int_{0}^{5} p^{2} \sin ϕ d p d θ d ϕ V = \int_{0}^{π} \sin ϕ d ϕ \int_{0}^{2 π} d θ [\int_{0}^{5} p^{2} d p] V = {[\frac{p^{3}}{3}]}_{0}^{5} \int_{0}^{π} \sin ϕ d ϕ \int_{0}^{2 π} d θ V = \frac{125}{3} \int_{0}^{π} \sin ϕ d ϕ \int_{0}^{2 π} d θ V = \frac{125}{3} (2 π) \int_{0}^{π} \sin ϕ d ϕ V = \frac{250}{3} π {[- \cos ϕ]}_{0}^{π} V = \frac{250}{3} π [1 + 1] V = \frac{500}{3} π cubic units$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Evaluating triple integrals: Each of the triple integrals that follows represents the volume of a solid. Sketch the solid and evaluate the integral.
$\int_{0}^{π} \int_{0}^{2 π} \int_{0}^{5} p^{2} \sin ϕ d p d θ d ϕ$

Short Answer

Step by step solution

Given

Evaluate integral

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Statistics

Logic and Functions

Pure Maths

Applied Mathematics

Probability and Statistics

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. ∫0π∫02π∫05p2sinϕdpdθdϕ

Short Answer

Step by step solution

Given

Evaluate integral

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Statistics

Logic and Functions

Pure Maths

Applied Mathematics

Probability and Statistics

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.
$\int_{0}^{π} \int_{0}^{2 π} \int_{0}^{5} p^{2} \sin ϕ d p d θ d ϕ$