/*! 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 23 Evaluate the double integral. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 23

Evaluate the double integral. $$ \int_{0}^{\infty} \int_{0}^{\infty} e^{-(x+y) / 2} d y d x $$

Short Answer

Expert verified
The value of the double integral is \(2\).

Step by step solution

01

Identify the type of integral

This is a double integral defined from \(0\) to \(\infty\). Each integral will have to be calculated separately, starting with the inner integral.
02

Evaluate the inner integral

The inner integral is: \(\int_{0}^{\infty} e^{-(x+y) / 2} d y\). As this integral is a function of \(y\), \(x\) is a constant. Apply the formula for calculating the integral of an exponential function which is, \(\int e^{-x} dx = -e^{-x} \), adding the limits of integral, from \(0\) to \( \infty\).
03

Solve the limit

Treat \(x\) as a constant and apply the limits to \(y\). This will give us \( -2 e^{-(x+y) / 2}\Big|_0^\infty \). Evaluate this, which will result in \( -2e^{-x}\). This is the result of the inner integral. Now, this expression is used for the second integral.
04

Compute the outer integral

The outer integral is \(\int_{0}^{\infty} -2e^{-x} dx\). Evaluate this integral using the same rules as in Step 2. After integrating, and applying the limits we have, \(-2(-1) = 2\)
05

Combine results

Combine these results. The value of the double integral is \(2\).

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.

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

Evaluate the double integral. $$ \int_{0}^{4} \int_{0}^{x} \frac{2}{x^{2}+1} d y d x $$

Use a double integral to find the area of the region bounded by the graphs of the equations. $$ y=9-x^{2}, y=0 $$

Evaluate the double integral. $$ \int_{1}^{2} \int_{0}^{4}\left(3 x^{2}-2 y^{2}+1\right) d x d y $$

Evaluate the double integral. $$ \int_{0}^{2} \int_{0}^{\sqrt{1-y^{2}}}-5 x y d x d y $$

Find the average value of \(f(x, y)\) over the region \(R\). $$ \begin{aligned} &f(x, y)=x y\\\ &R: \text { rectangle with vertices }(0,0),(4,0),(4,2),(0,2) \end{aligned} $$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

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.