# Double Integral with Strange Limits of Integration

## Homework Statement

For the double integral ∫[0,1]∫[0,x^3] e^(y/x) dxdy
(a) sketch the region of integration
(b) evaluate the integral and
(c) re-express the integral with the order of integration reversed

None

## The Attempt at a Solution

The problem is that I've never seen a double integral problem with the limits of integration with respect to x in terms of a function of x, not y. I couldn't find any examples online or in my book where this is the case. The problem is written such that your boundaries should be 0≤x≤x^3 and 0≤y≤1... but x=x^3 doesn't make any sense to me as an upper boundary for x.

I know how to do most double integral problems, reverse the order of integration, etc. but this has me stumped. Am I right in thinking this might be a typo? or is there some way to make sense of the region that I'm just not seeing?

Related Calculus and Beyond Homework Help News on Phys.org
DryRun
Gold Member
Hi bossman27 and welcome to PF!

Using LaTeX, here is how your double integral appears:
$$\int^1_0 \int^{x^3}_0 e^{(y/x)}\,.dxdy$$
But it should have been:
$$\int^{x=1}_{x=0} \int^{y=x^3}_{y=0} e^{(y/x)}\,.dydx$$

Mark44
Mentor

## Homework Statement

For the double integral ∫[0,1]∫[0,x^3] e^(y/x) dxdy
(a) sketch the region of integration
(b) evaluate the integral and
(c) re-express the integral with the order of integration reversed

None

## The Attempt at a Solution

The problem is that I've never seen a double integral problem with the limits of integration with respect to x in terms of a function of x, not y. I couldn't find any examples online or in my book where this is the case. The problem is written such that your boundaries should be 0≤x≤x^3 and 0≤y≤1... but x=x^3 doesn't make any sense to me as an upper boundary for x.

I know how to do most double integral problems, reverse the order of integration, etc. but this has me stumped. Am I right in thinking this might be a typo? or is there some way to make sense of the region that I'm just not seeing?
0 <= x <= x3 doesn't make sense to me either. That should be 0 <= y <= x3, and 0 <=x <= 1.

Are you sure you don't have dx and dy switched?
This would make more sense.
$$\int_{x = 0}^1\int_{y = 0}^{x^3}e^{y/x}dy~dx$$

0 <= x <= x3

Are you sure you don't have dx and dy switched?
This would make more sense.
$$\int_{x = 0}^1\int_{y = 0}^{x^3}e^{y/x}dy~dx$$
I'm sure I did not switch them, that was exactly how the problem appears on this practice test... I was thinking that it was probably a typo as well, but as the final is tomorrow morning I wanted to make sure I wasn't missing something.

Thanks to both of you for the quick responses!

It's entirely possible that you are essentially being asked for the anti-derivative, so you should think of the bound for the integral as a parameter, and the internal integral as being taken with respect to some dummy variable $x^\prime$.

Mark44
Mentor
It's entirely possible that you are essentially being asked for the anti-derivative, so you should think of the bound for the integral as a parameter, and the internal integral as being taken with respect to some dummy variable $x^\prime$.
That's an interesting idea, but I don't think it's applicable here. bossman27 said that the integral was exactly as it appeared on the practice test. I believe that the instructor switched the order of dx and dy by mistake.