/*! 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 59 Reverse the order of integration... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 59

Reverse the order of integration in the following integrals. $$\int_{1 / 2}^{1} \int_{0}^{-\ln y} f(x, y) d x d y$$

Short Answer

Expert verified
The new integral, after reversing the order of integration, is $\int_{0}^{\ln 2} \int_{1}^{e^{-x}} f(x, y) dy dx$.

Step by step solution

01

Identify the region of integration

First, let's find the region of integration in the xy-plane. We can do this by using the bounds of integration in the given integral. For \(y\), we can see that it ranges from \(1/2\) to \(1\). For \(x\), it ranges from \(0\) to \(-\ln y\). Since we have a logarithmic function, we need to find the equation for this curve to be able to reverse the order of integration. The curve is given by \(x = -\ln y\) or \(-\ln y = x\).
02

Rewrite the constraint in an appropriate form

Since we want to rewrite the integral with respect to \(dy\), it is better to have the constraint in terms of \(y\). So, we need to rewrite the curve as a function of \(x\). We can do this by expressing \(y\) in terms of \(x\). From \(x = -\ln y\), we can write: $$y = e^{-x}$$
03

Determine the new bounds for x and y

Now, find the intersection points of the curve \(y = e^{-x}\) with the vertical bounds \(y = 1/2\) and \(y = 1\). When \(y = 1\): $$1 = e^{-x} \Rightarrow x = 0$$ When \(y = 1/2\): $$\frac{1}{2} = e^{-x} \Rightarrow x = \ln 2$$ So we have \(x\) ranging from \(0\) to \(\ln 2\). For \(y\), it will range from its minimum value \(y=1\) at \(x=0\) to its maximum value \(y=1/2\) at \(x=-\ln y\) or \(x = \ln 2\).
04

Rewrite the integral

Now we rewrite the original integral with the reversed order of integration, using the new bounds for \(x\) and \(y\) determined in step 3. The new integral is: $$\int_{0}^{\ln 2} \int_{1}^{e^{-x}} f(x, y) dy dx$$

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

Mass from density Find the mass of the following solids with the given density functions. Note that density is described by the function \(f\) to avoid confusion with the radial spherical coordinate \(\rho\). The solid cylinder \(\\{(r, \theta, z): 0 \leq r \leq 2,0 \leq \theta \leq 2 \pi\) \(-1 \leq z \leq 1\\}\) with a density of \(f(r, \theta, z)=(2-|z|)(4-r)\)

The following table gives the density (in units of \(\mathrm{g} / \mathrm{cm}^{2}\) ) at selected points of a thin semicircular plate of radius 3. Estimate the mass of the plate and explain your method. $$\begin{array}{|c|c|c|c|c|c|} \hline & \boldsymbol{\theta}=\mathbf{0} & \boldsymbol{\theta}=\boldsymbol{\pi} / \boldsymbol{4} & \boldsymbol{\theta}=\boldsymbol{\pi} / \boldsymbol{2} & \boldsymbol{\theta}=\boldsymbol{3} \pi / \boldsymbol{4} & \boldsymbol{\theta}=\boldsymbol{\pi} \\ \hline \boldsymbol{r}=\mathbf{1} & 2.0 & 2.1 & 2.2 & 2.3 & 2.4 \\ \hline \boldsymbol{r}=\mathbf{2} & 2.5 & 2.7 & 2.9 & 3.1 & 3.3 \\ \hline \boldsymbol{r}=\mathbf{3} & 3.2 & 3.4 & 3.5 & 3.6 & 3.7 \\ \hline \end{array}$$

Let \(R\) be the region bounded by the ellipse \(x^{2} / a^{2}+y^{2} / b^{2}=1,\) where \(a>0\) and \(b>0\) are real numbers. Let \(T\) be the transformation \(x=a u, y=b v\) Find the average square of the distance between points of \(R\) and the origin.

Use spherical coordinates to find the volume of the following solids. The solid cardioid of revolution \(D=\\{(\rho, \varphi, \theta): 0 \leq \rho \leq 1+\cos \varphi, 0 \leq \varphi \leq \pi, 0 \leq \theta \leq 2 \pi\\}\)

Let \(R\) be the region bounded by the ellipse \(x^{2} / a^{2}+y^{2} / b^{2}=1,\) where \(a>0\) and \(b>0\) are real numbers. Let \(T\) be the transformation \(x=a u, y=b v\) Evaluate \(\iint_{R}|x y| d A\)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

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.