15.2.87 Write the following integrals as a single iterated integral.

In summary, the single iterated integral is $\displaystyle\int_{0}^{1} \int_{e^y}^{e} f(x,y)\,dx\,dy + \int_{-1}^{0} \int_{e^{-y}}^{e}f(x,y) \,dx\,dy$
  • #1
karush
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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
 
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  • #2
Re: 15.2.87 write the following integrals as a single iterated integral.

karush said:
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:

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

We see that this region is:

\(\displaystyle e^y\le x\le e\)

\(\displaystyle 0\le y\le1\)

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

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

We see that this region is:

\(\displaystyle e^{-y}\le x\le e\)

\(\displaystyle -1\le y\le0\)

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 $y$?
 

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  • #3
Re: 15.2.87 write the following integrals as a single iterated integral.

the vertical strips in terms of y would be
$$-1\le y \le 1$$
 
  • #4
Re: 15.2.87 write the following integrals as a single iterated integral.

karush said:
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.
 
  • #5
Re: 15.2.87 write the following integrals as a single iterated integral.

karush said:
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?
 
  • #6
Re: 15.2.87 write the following integrals as a single iterated integral.

so you mean

$1 \le x \le e $
 
  • #7
Re: 15.2.87 write the following integrals as a single iterated integral.

karush said:
so you mean

$1 \le x \le e $

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

\(\displaystyle x=e^{-y}\)

Now, we want to solve this for $y$...what do you get?
 
  • #8
Re: 15.2.87 write the following integrals as a single iterated integral.

$\ln {x} = -y$
$-\ln{x} =y$
 
  • #9
Re: 15.2.87 write the following integrals as a single iterated integral.

karush said:
$\ln {x} = -y$
$-\ln{x} =y$

Okay, good...how about the upper limit?
 
  • #10
Re: 15.2.87 write the following integrals as a single iterated integral.

MarkFL said:
Okay, good...how about the upper limit?

I would presume simply
$\ln{x}=y$
 
  • #11
Re: 15.2.87 write the following integrals as a single iterated integral.

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

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

karush said:
so you mean

$1 \le x \le e $

Putting everything together, what is the single iterated integral?
 

Related to 15.2.87 Write the following integrals as a single iterated integral.

1. What is an iterated integral?

An iterated integral is a way of solving a double or triple integral by breaking it down into multiple single integrals. This allows for a more manageable and easier-to-solve integral.

2. How do you convert multiple integrals into a single iterated integral?

To convert multiple integrals into a single iterated integral, you must first identify the limits of integration for each variable. Then, you can rewrite the integral using the limits of integration for each variable, with the innermost integral being the first variable and the outermost integral being the last variable.

3. What is the purpose of writing an integral as a single iterated integral?

Writing an integral as a single iterated integral allows for an easier method of solving the integral, as well as providing a better understanding of the function being integrated.

4. How do you know which variable to integrate with first in a single iterated integral?

The variable that is integrated first in a single iterated integral is the innermost integral. This is typically the variable with the smallest range of integration.

5. Can any integral be written as a single iterated integral?

Yes, any double or triple integral can be written as a single iterated integral. However, the process may become more complex for higher order integrals.

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