/*! 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 9 Set up a triple integral for the... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 9

Set up a triple integral for the volume of the solid. The solid bounded by the paraboloid \(z=9-x^{2}-y^{2}\) and the plane \(z=0\)

Short Answer

Expert verified
The volume of the solid bounded by the paraboloid \(z=9-x^{2}-y^{2}\) and the plane \(z=0\) can be computed by the triple integral \(V=\int_{0}^{2\pi} \int_{0}^{3} \int_{0}^{9-r^{2}} rdrd\theta dz \)

Step by step solution

01

Understanding the solid

The first step is to understand the shapes of the paraboloid \(z=9-x^{2}-y^{2}\) and the plane \(z=0\). Here, \(z=9-x^{2}-y^{2}\) represents a paraboloid that opens downward, with its vertex at the point (0,0,9), and \(z=0\) is a flat plane on the xy-plane.
02

Setting up the boundaries

To integrate the solid bounded by these two surfaces, cylindrical coordinates are more efficient due to the symmetry of the paraboloid. In cylindrical coordinates \((r, \theta, z)\), \(x^{2}+y^{2} = r^{2}\). So the equation of the paraboloid becomes \(z=9-r^{2}\). The xy projection of the solid is a circle, thus \(0 \leq r \leq 3\) and \(0 \leq \theta \leq 2\pi\). For z, the solid lies between the plane \(z=0\) and the paraboloid \(z=9-r^{2}\). Therefore, \(0 \leq z \leq 9-r^{2}\).
03

Setting up the triple integral

Now it's time to set up the triple integral for the volume. The volume element in cylindrical coordinates is \(dV=rdrd\theta dz\). Thus, the volume of the solid is given by the triple integral \(V=\int\int\int dV=\int_{0}^{2\pi} \int_{0}^{3} \int_{0}^{9-r^{2}} rdrd\theta dz \)

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

Determine which value best approximates the volume of the solid between the \(x y\) -plane and the function over the region. (Make your selection on the basis of a sketch of the solid and not by performing any calculations.) \(f(x, y)=x y+2 ; R:\) quarter circle: \(x^{2}+y^{2}=9, x \geq 0, y \geq 0\) (a) 25 (b) 8 (c) 100 (d) 50 (e) -30

In Exercises \(51-54,\) evaluate the iterated integral. (Note that it is necessary to switch the order of integration.) $$ \int_{0}^{2} \int_{x}^{2} e^{-y^{2}} d y d x $$

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.) $$ x y=4, x=1, x=4, \rho=k x^{2} $$

In Exercises \(37-42,\) sketch the region \(R\) of integration and switch the order of integration. $$ \int_{0}^{4} \int_{0}^{y} f(x, y) d x d y $$

In Exercises 7 and 8, convert the integral from rectangular coordinates to both cylindrical and spherical coordinates, and evaluate the simplest iterated integral. $$ \int_{-2}^{2} \int_{-\sqrt{4-x^{2}}}^{\sqrt{4-x^{2}}} \int_{x^{2}+y^{2}}^{4} x d z d y d x $$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

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.