MHB 15.2.87 Write the following integrals as a single iterated integral.

[karush]

Draw the regions of integration and write the following integrals as a single iterated integral.
$$\displaystyle\int_{0}^{1} \int_{e^y}^{e} f(x,y)\,dx\,dy + \int_{-1}^{0} \int_{e^{-y}}^{e}f(x,y) \,dx\,dy$$
ok haven't done this before so kinda clueless

 

[MarkFL]

Re: 15.2.87 write the following integrals as a single iterated integral.

Draw the regions of integration and write the following integrals as a single iterated integral.
$$\displaystyle\int_{0}^{1} \int_{e^y}^{e} f(x,y)\,dx\,dy + \int_{-1}^{0} \int_{e^{-y}}^{e}f(x,y) \,dx\,dy$$
ok haven't done this before so kinda clueless

Let's begin with the first integral:

$$I_1=\int_{0}^{1} \int_{e^y}^{e} f(x,y)\,dx\,dy$$

We see that this region is:

$$e^y\le x\le e$$

$$0\le y\le1$$

We'll fill in that region with red. For the second integral, we have:

$$I_2=\int_{-1}^{0} \int_{e^{-y}}^{e}f(x,y) \,dx\,dy$$

We see that this region is:

$$e^{-y}\le x\le e$$

$$-1\le y\le0$$

We'll fill that region in with green...so we have:

View attachment 7255

In order to write this as a single integral, we'll need to reverse the order of integration, and use vertical strips. Can you state the lower and upper limits for these vertical strips in terms of $y$?

 

[karush]

Re: 15.2.87 write the following integrals as a single iterated integral.

the vertical strips in terms of y would be
$$-1\le y \le 1$$

 

[Prove It]

Re: 15.2.87 write the following integrals as a single iterated integral.

the vertical strips in terms of y would be
$$-1\le y \le 1$$

Not even close. Draw in a vertical strip ANYWHERE in the region. It should be obvious that depending on where you draw it, the strip will have VARIABLE length.

 

[MarkFL]

Re: 15.2.87 write the following integrals as a single iterated integral.

the vertical strips in terms of y would be
$$-1\le y \le 1$$

I was kind of sloppy in my language before...you want the $y$-coordinates of the lower and upper ends of the strips (since the inner integral has the differential $dy$), but you want them in terms of $x$.

Does that make sense?

 

[karush]

Re: 15.2.87 write the following integrals as a single iterated integral.

so you mean

$1 \le x \le e $

 

[MarkFL]

Re: 15.2.87 write the following integrals as a single iterated integral.

so you mean

$1 \le x \le e $

No, the bottom of a vertical strip is on the curve:

$$x=e^{-y}$$

Now, we want to solve this for $y$...what do you get?

 

[karush]

Re: 15.2.87 write the following integrals as a single iterated integral.

$\ln {x} = -y$
$-\ln{x} =y$

 

[MarkFL]

Re: 15.2.87 write the following integrals as a single iterated integral.

$\ln {x} = -y$
$-\ln{x} =y$

Okay, good...how about the upper limit?

 

[karush]

Re: 15.2.87 write the following integrals as a single iterated integral.

Okay, good...how about the upper limit?

I would presume simply
$\ln{x}=y$

 

[MarkFL]

Re: 15.2.87 write the following integrals as a single iterated integral.

I would presume simply
$\ln{x}=y$

Yes, and for the outer integral, you've already correctly stated:

so you mean

$1 \le x \le e $

Putting everything together, what is the single iterated integral?