Q. 29 - Calculus

Question

In Exercises 29–34, sketch the region determined by the limits of the iterated integrals and then give another iterated integral (or a sum of iterated integrals if necessary) using the opposite order of integration. 

${\int }_1^2{\int }_{\ln x}^{e^x}f(x,y)dydx$

Step-by-Step Solution

Answer

The sketch of the region is;

The iterated integral:

${\int }_0^1{\int }_1^{e^y}f(x,y)dxdy+{\int }_1^e{\int }_1^{2}f(x,y)dxdy+{\int }_e^{e^2}{\int }_{\ln y}^{2}f(x,y)dxdy$

Step 1. Given information

Integral:

${\int }_1^2{\int }_{\ln x}^{e^x}f(x,y)dydx$

Step 2. Sketch the region:

In the given integral we can see that the region is :

$$\begin{gathered} \ln x\le y\le e^x \\ 1\le x\le 2 \end{gathered}$$

So the sketch is :

Step 3. Write another iterated integral.

When $y=\ln x$ then $x=e^y$

When $y=e^x$ then $x=\ln y$

The region is divided into three parts:

When $x=1$ then $$\begin{gathered} y=\ln 1 \\ =0 \end{gathered}$$

When $x=1$ then $$\begin{gathered} y=e^1 \\ =e \end{gathered}$$

When $x=2$ then $y=\ln 2,e^2$

The integral can be changed as:

${\int }_0^1{\int }_1^{e^y}f(x,y)dxdy+{\int }_1^e{\int }_1^{2}f(x,y)dxdy+{\int }_e^{e^2}{\int }_{\ln y}^{2}f(x,y)dxdy$