Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 5: Problem 20

(a) Find the area of the surface \(z=1+x^{2}+y^{2}\) inside the cylinder \(x^{2}+y^{2}=1\). (b) Find the volume inside the cylinder between the surface and the \((x, y)\) plane. Use cylindrical coordinates.

Short Answer

Expert verified
Area \(= \frac{\pi}{6} (5^{3/2} - 1) \), Volume \(= \frac{3\pi}{2} \).

Step by step solution

01

Understand the problem

(a) Area of surface under z = 1 + x² + y² inside the cylinder x² + y² = 1. (b) Volume inside the cylinder x² + y² = 1, between the surface and (x, y) plane.
02

Convert to cylindrical coordinates

In cylindrical coordinates, use: \[ x = r \cos \theta, \; y = r \sin \theta, \; z = z \]Here, \( x^2 + y^2 = r^2 \). So the surface becomes \( z = 1 + r^2 \) and the boundary is \( r = 1 \).
03

Compute surface area integral

The formula for the surface area is \[ A = \iint_{D} \sqrt{1+\left(\frac{\partial z}{\partial x}\right)^2 + \left(\frac{\partial z}{\partial y}\right)^2} \; dA \]Converting to cylindrical coordinates and simplifying:\[ \iint_{D} \sqrt{1 + 4r^2} \; r \; dr \; d\theta \]The limits are from 0 to 2π for \( \theta \) and 0 to 1 for \( r \).Therefore, the integral becomes: \[ \int_{0}^{2\pi} \int_{0}^{1} \sqrt{1+4r^2} \; r \; dr \; d\theta \]
04

Perform the integral

Solve the integral for \( r \): \[ \int_{0}^{1} r \sqrt{1+4r^2} \; dr \]Set \( u = 1 + 4r^2 \), hence \( du = 8r \; dr \) and \( dr = \frac{du}{8r} \), yielding:\[ \int \frac{1}{8} u^{1/2} du \]Thus, it becomes: \[ \frac{1}{12} \left[ u^{3/2} \right]_{1}^{5} = \frac{1}{12} \left[ (1+ 4 \times 1)^{3/2} - 1\right] = \frac{1}{12} (5^{3/2} - 1) \]Multiply this by the integral over \( \theta \) to get the surface area.\[ 2\pi \times \frac{1}{12} \left( 5^{3/2} - 1 \right) \]
05

Compute volume integral

The volume inside the cylinder, bounded by \( z = 1 + r^2 \) and the plane, is given by:\[ V = \iint_{R} \ (1+r^2) \, r \, dr \, d\theta \]Following similar limits, we get:\[ V = \int_{0}^{2\pi} \int_{0}^{1} (1+r^2) \, r \, dr \, d\theta \]
06

Perform the volume integral

Separate the integral into two parts:\[ \int_{0}^{1} r \, dr + \int_{0}^{1} r^3 \, dr \]Which evaluates to: \[ \left. \frac{r^2}{2} \right|_{0}^{1} + \left. \frac{r^4}{4} \right|_{0}^{1} = \frac{1}{2} + \frac{1}{4} = \frac{3}{4} \]Then multiplying by the \( \theta \) integral:\[ 2\pi \times \frac{3}{4} = \frac{3\pi}{2} \]

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

cylindrical coordinates
Cylindrical coordinates are a three-dimensional coordinate system that extends polar coordinates by adding a height dimension. Instead of using Cartesian coordinates \( x \), \( y \), and \( z \), cylindrical coordinates are represented as \( (r, \theta, z) \). Here, \( r \) is the radial distance from the origin in the \( xy \)-plane, \( \theta \) is the angular position around the \( z \)-axis, and \( z \) is the height above the \( xy \)-plane.
Using cylindrical coordinates can significantly simplify certain types of integrals, especially when dealing with problems that have radial symmetry. In our example, we convert \( x \) and \( y \) to \( r \) and \( \theta \) for easier calculation. We have:
  • \

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

(a) Write a triple integral in spherical coordinates for the volume inside the cone \(z^{2}=x^{2}+y^{2}\) and between the planes \(z=1\) and \(z=2\). Evaluate the integral. (b) Do (a) in cylindrical coordinates.

For a square lamina of uniform density find \(I\) about (a) a side, (b) a diagonal, (c) an axis through a corner and perpendicular to its plane. Hint: See the perpendicular axis theorem,

Evaluate the double integrals over the areas described. To find the limits, sketch the area and compare $$ \iint_{A}(2 x-3 y) d x d y \text {, where } A \text { is the triangle with vertices }(0,0),(2,1),(2,0) \text {. } $$

Evaluate the double integrals over the areas described. To find the limits, sketch the area and compare \(\iint 2 x y d x d y\) over the triangle with vertices \((0,0),(2,1),(3,0)\).

Find the volume between the planes \(z=2 x+3 y+6\) and \(z=2 x+7 y+8\) and over the square in the \((x, y)\) plane with vertices \((0,0),(1,0),(0,1),(1,1)\).
See all solutions

Recommended explanations on Combined Science Textbooks

Synergy

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.

Sign-up for free