/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 10 Set up a triple integral for the... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 10

Set up a triple integral for the volume of the solid. The solid that is the common interior below the sphere \(x^{2}+y^{2}+z^{2}=80\) and above the paraboloid \(z=\frac{1}{2}\left(x^{2}+y^{2}\right)\)

Short Answer

Expert verified
The triple integral that represents the volume of the solid within the sphere and above the paraboloid is \(\int_0^{2蟺}\int_0^{arccos\sqrt{1 - 蟻^{2}}}\int_0^{\sqrt{80}} 蟻^{2}sin(蠁)d蟻d蠁d胃\).

Step by step solution

01

Convert the given equations into spherical coordinates

In spherical coordinates, the sphere \(x^{2}+y^{2}+z^{2}=80\) becomes \(蟻^{2}=80\) or \(蟻= \sqrt{80}\). The paraboloid \(z=\frac{1}{2}\left(x^{2}+y^{2}\right)\) becomes \(\cos(蠁)=蟻sin(蠁)\) or \(蠁=\arccos\sqrt{1 - 蟻^{2}}\). The angle \(胃\) will range over all values, which is [0, 2蟺]. The volume element in spherical coordinates is \(dV = 蟻^{2}sin(蠁)d蟻d蠁d胃\).
02

Set up the integral

Now we set up the triple integral. The limits of the integral for \(蠁\) are determined by the sphere and the paraboloid, which are: \([0, \arccos\sqrt{1 - 蟻^{2}}]\). The limits for \(胃\) are [0, 2蟺] as \(胃\) covers all the angles in a circle around the positive z-axis. The limits for \(蟻\) are [0, \sqrt{80}] as \(蟻\) ranges from the origin to surface of the sphere. We integrate the volume element \(\int^{\sqrt{80}}_\sigma \int^{2蟺}_0 \int^{arccos\sqrt{1 - 蟻^{2}}} _0 蟻^{2}sin(蠁)d蠁d胃d蟻\).
03

Simplify the integral

Simplify the triple integral: in this problem we don't actually carry out the integration because the task only asks for setting up the integral for the volume of the solid.

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the mass and center of mass of the lamina bounded by the graphs of the equations for the given density or densities. (Hint: Some of the integrals are simpler in polar coordinates.) \(x^{2}+y^{2}=a^{2}, 0 \leq x, 0 \leq y\) (a) \(\rho=k\) (b) \(\rho=k\left(x^{2}+y^{2}\right)\)

In Exercises \(43-50\), sketch the region \(R\) whose area is given by the iterated integral. Then switch the order of integration and show that both orders yield the same area. $$ \int_{-2}^{2} \int_{-\sqrt{4-x^{2}}}^{\sqrt{4-x^{2}}} d y d x $$

In Exercises \(1-10\), evaluate the integral. $$ \int_{e^{y}}^{y} \frac{y \ln x}{x} d x, \quad y>0 $$

In Exercises \(11-22,\) evaluate the iterated integral. $$ \int_{0}^{2} \int_{y}^{2 y}\left(10+2 x^{2}+2 y^{2}\right) d x d y $$

In Exercises \(43-50\), sketch the region \(R\) whose area is given by the iterated integral. Then switch the order of integration and show that both orders yield the same area. $$ \int_{-2}^{2} \int_{0}^{4-y^{2}} d x d y $$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.