Sign Error in Double Integral Word Problem

[burt]

I am working on the following problem:

I worked through the problem, but I seem to have made a sign error along the way. From what I can see, it seems that if I did make a sign error it was as I completed the integration with respect to x. But, I can't figure out what I did wrong! Any ideas?
Here is my work:


Thank you!

 

[Steven G]

[math]\int e^{-x^2}dx \neq \dfrac{e^{-x^2}}{2}[/math]

 

[burt]

[math]\int e^{-x^2}dx \neq \dfrac{e^{-x^2}}{2}[/math]

But it is \(\int xe^{-x^2}dx\) not just \(\int e^{-x^2} dx \)

 

[burt]

I did \(u=x^2\) and \(\frac{du}{dx}=2x\) so I ended up with \(\int \frac{e^{-u}}{2}du\)
But I do see my sign error - the solution to that integral is negative, not positive.

 

[Steven G]

I did \(u=x^2\) and \(\frac{du}{dx}=2x\) so I ended up with \(\int \frac{e^{-u}}{2}du\)
But I do see my sign error - the solution to that integral is negative, not positive.

You need to replace everything with x's into u's. You can have nothing left over!
[math]\int e^{-x^2}dx[/math] u=x^2 and du = 2xdx or du/(2x) = dx.[/math] Then [math]\int e^{-x^2}dx[/math] becomes [math]\int e^{-u} \frac {du}{2x} =\int e^{-u} \frac {du}{2\sqrt{u}}[/math]
You lost the x in the denominator!!!!!

 

[Steven G]

I did \(u=x^2\) and \(\frac{du}{dx}=2x\) so I ended up with \(\int \frac{e^{-u}}{2}du\)
But I do see my sign error - the solution to that integral is negative, not positive.

Yes, du/dx = 2x. So how does this change the integral when you make your substitution???

 

[burt]

Yes, du/dx = 2x. So how does this change the integral when you make your substitution???

yes, but there is an x in the problem - so i had \(\frac{du}{2}=xdx\) and I took out both the x and the dx.

 

[Steven G]

There was? If that is the case can you please state the correct integral, the one with the x in it.

 

[Steven G]

But it is \(\int xe^{-x^2}dx\) not just \(\int e^{-x^2} dx \)

ok, I see the x now. But where did it come from?

 

[burt]

There was? If that is the case can you please state the correct integral, the one with the x in it.

The one originally posted is correct.
I'll copy it here:

 

[burt]

ok, I see the x now. But where did it come from?

The original problem.

 

[Steven G]

Oh sorry! Now I see that x in the original integral. Sorry about all this!

 

[Steven G]

Again I apologize for this delay. You did lose a negative sign. Can you try again and see if you get the correct results?

 

[burt]

Again I apologize for this delay. You did lose a negative sign. Can you try again and see if you get the correct results?

Yup! I think I found my mistake. Thank you so much!