/*! 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 12 Find the area of the surface. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 12

Find the area of the surface. The portion of the cone \(z=2 \sqrt{x^{2}+y^{2}}\) inside the cylinder \(x^{2}+y^{2}=4\)

Short Answer

Expert verified
The area of the surface defined by the portion of the cone \(z=2 \sqrt{x^{2}+y^{2}}\) inside the cylinder \(x^{2}+y^{2}=4\) is \(A = \frac{64}{3}\pi\).\n

Step by step solution

01

Identify the Parameters of the Surface

We first need to identify the surface in question. The surface is defined by the function \(z = 2 \sqrt{x^{2}+y^{2}}\) which represents a cone. We are looking specifically at the portion of the cone inside a cylinder, which is defined by \(x^{2} + y^{2} = 4\). This gives us a radius of 2 for our surface.
02

Parameterize the Surface

With our radius known, we can now parameterize our surface in terms of cylindrical coordinates (r, 胃, z). Given \(x = r \cos(胃), y = r \sin(胃)\), we can substitute these into the equation for our cone to obtain: \(z = 2 r\). The boundaries of \(r\) and \(胃\) are determined by the cylinder: \(0 鈮� r 鈮� 2\), and \(0 鈮� 胃 鈮� 2蟺\). Now, find the differential surface area in cylindrical coordinates, which is \(rdrd胃\).
03

Implement Surface Area Integral

The formula for the surface area of a parameterized surface like ours is given by the double integral \[A = \int \int \sqrt{1 + (f_{x})^{2} + (f_{y})^{2}} \, dS\] where \(f_{x}\) and \(f_{y}\) are the partial derivatives of \(z\) with respect to \(x\) and \(y\) respectively, and \(dS = rdrd胃\) is the differential surface area.
04

Compute the Surface Area Integral

To apply the formula, first calculate the partial derivatives, then compute the square root term, which with simplification will equal \(\sqrt{r^{2} + 4}\). Then integrate over \(r\) and \(胃\): \[A = \int_0^{2\pi} \int_0^2 r \sqrt{r^{2} + 4} \, dr\,d胃.\] This integral can be solved iteratively using standard integration techniques, such as u-substitution.
05

Final Calculation

The integral over r returns 32/3, which can then be multiplied by the integral over 胃, resulting in the final surface area of the portion of the cone inside the cylinder as \(A = \frac{64}{3}\pi\).

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

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 \(43-50\), 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. $$ \int_{0}^{1} \int_{-\sqrt{1-y^{2}}}^{\sqrt{1-y^{2}}} d x d y $$

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

Prove the following Theorem of Pappus: Let \(R\) be a region in a plane and let \(L\) be a line in the same plane such that \(L\) does not intersect the interior of \(R .\) If \(r\) is the distance between the centroid of \(R\) and the line, then the volume \(V\) of the solid of revolution formed by revolving \(R\) about the line is given by \(V=2 \pi r A,\) where \(A\) is the area of \(R\)

True or False? In Exercises 65 and \(66,\) determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \int_{a}^{b} \int_{c}^{d} f(x, y) d y d x=\int_{c}^{d} \int_{a}^{b} f(x, y) d x d y $$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Pure Maths

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.