# Integration/Differentiation help.

• Nov 15th 2011, 11:23 AM
Wevans2303
Integration/Differentiation help.
Can someone please explain how I would differentiate this:

(1-e^(-x^2))(1-e^(-y^2))

and integrate this:

24y(1-x-y)

Regards.
• Nov 15th 2011, 12:09 PM
TheEmptySet
Re: Integration/Differentiation help.
Quote:

Originally Posted by Wevans2303
Can someone please explain how I would differentiate this:

(1-e^(-x^2))(1-e^(-y^2))

and integrate this:

24y(1-x-y)

Regards.

You need to be more specific.

For the first one are you taking partial derivatives or is y a function of x?

For the integral what are you integrating with respect to? dx, dy, dA, ds ? please clarify.

• Nov 15th 2011, 12:10 PM
Wevans2303
Re: Integration/Differentiation help.
Apologies TES.

Differentiating w.r.t x and a double integral dydx.
• Nov 15th 2011, 12:26 PM
TheEmptySet
Re: Integration/Differentiation help.
So for the first question if you have

$f(x,y)=\dfrac{1-e^{-x^2}}{1-e^{-y^2}}$

The partial derivative with respect to x is

$\dfrac{\partial f}{\partial x}=\dfrac{1}{1-e^{-y^2}}\dfrac{\partial }{\partial x} (1-e^{-x^2})= \dfrac{1}{1-e^{-y^2}}(0-e^{-x^2}(-2x))=\dfrac{2xe^{-x^2}}{1-e^{-y^2}}$

For the 2nd one we need to know the domain of integration. What are the limits?
• Nov 15th 2011, 12:46 PM
Wevans2303
Re: Integration/Differentiation help.
Oh god i've totally messed this up.

It's the other way round, integrate the first differentiate the second, sorry to have wasted your time.

If you're still in the mood to help, it would be the double integral of 24y(1-x-y ) with limits 0,1 for both x and y.
• Nov 15th 2011, 01:01 PM
tom@ballooncalculus
Re: Integration/Differentiation help.
Quote:

Originally Posted by TheEmptySet
So for the first question if you have
$f(x,y)=\dfrac{1-e^{-x^2}}{1-e^{-y^2}}$

... whereas, if y is a function of x...

http://www.ballooncalculus.org/draw/diffProd/five.png

... where (key in spoiler) ...

Spoiler:
http://www.ballooncalculus.org/asy/chain.png

... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to the main variable (in this case x), and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).
But this is wrapped inside each leg of the legs-uncrossed version of...

http://www.ballooncalculus.org/asy/prod.png

... the product rule, where, again, straight continuous lines are differentiating downwards with respect to x.

The second...

http://www.ballooncalculus.org/draw/diffProd/fivea.png

I'll maybe change it to use the limits.

Here we are, then...

http://www.ballooncalculus.org/draw/diffProd/fiveb.png

Quote:

Originally Posted by Wevans2303
Oh god i've totally messed this up.

It's the other way round, integrate the first differentiate the second, sorry to have wasted your time.

If you're still in the mood to help, it would be the double integral of 24y(1-x-y ) with limits 0,1 for both x and y.

But that would still be integrating the second?!
__________________________________________________ __________

Don't integrate - balloontegrate!

Balloon Calculus; standard integrals, derivatives and methods

Balloon Calculus Drawing with LaTeX and Asymptote!
• Nov 15th 2011, 01:40 PM
Wevans2303
Re: Integration/Differentiation help.
Quote:

Originally Posted by tom@ballooncalculus
... whereas, if y is a function of x...

http://www.ballooncalculus.org/draw/diffProd/five.png

... where (key in spoiler) ...

Spoiler:
http://www.ballooncalculus.org/asy/chain.png

... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to the main variable (in this case x), and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).
But this is wrapped inside each leg of the legs-uncrossed version of...

http://www.ballooncalculus.org/asy/prod.png

... the product rule, where, again, straight continuous lines are differentiating downwards with respect to x.

The second...

http://www.ballooncalculus.org/draw/diffProd/fivea.png

I'll maybe change it to use the limits.

Here we are, then...

http://www.ballooncalculus.org/draw/diffProd/fiveb.png

But that would still be integrating the second?!
__________________________________________________ __________

Don't integrate - balloontegrate!

Balloon Calculus; standard integrals, derivatives and methods

Balloon Calculus Drawing with LaTeX and Asymptote!

I'm not making a good first impression here am I!

Sorry, I need help with integrating the first one with limits 0,1 for both x,y and differentiating the second one.
• Nov 15th 2011, 02:26 PM
tom@ballooncalculus
Re: Integration/Differentiation help.
Ok, and is your first expression a fraction as TES was guessing? Either way, this looks dodgy (i.e. non-elementary, involving the error function).

And in the second, is y a function of x?
• Nov 15th 2011, 03:08 PM
Wevans2303
Re: Integration/Differentiation help.
Quote:

Originally Posted by tom@ballooncalculus
Ok, and is your first expression a fraction as TES was guessing? Either way, this looks dodgy (i.e. non-elementary, involving the error function).

And in the second, is y a function of x?

I need to integrate this (double integration where limits of x are 0,1 and y are 0,1):

f(x,y) = 1-e^(-x^2)* 1-e^(-y^2) for x >0, y>0

I need to differentiate this:

f(x,y) = 24y(1-x-y) x>0,y>0 & x+y<1

That's all I am given in the questions.
• Nov 16th 2011, 01:48 AM
tom@ballooncalculus
Re: Integration/Differentiation help.
Quote:

Originally Posted by Wevans2303
I need to integrate this (double integration where limits of x are 0,1 and y are 0,1):

f(x,y) = 1-e^(-x^2)* 1-e^(-y^2) for x >0, y>0

int 0 to 1 int 0 to 1 &#40;1-e&#94;&#40;-x&#94;2&#41;&#41;&#40;1-e&#94;&#40;-y&#94;2&#41;&#41; dy dx - Wolfram|Alpha

I.e...

http://www.ballooncalculus.org/draw/double/four.png

To differentiate f(x.y) with respect to (e.g.) x, expand and differentiate term by term, treating y as constant.

__________________________________________________ ____________________

Don't integrate - balloontegrate!

Balloon Calculus; standard integrals, derivatives and methods

Balloon Calculus Drawing with LaTeX and Asymptote!
• Nov 16th 2011, 03:45 AM
Wevans2303
Re: Integration/Differentiation help.
Quote:

Originally Posted by tom@ballooncalculus
int 0 to 1 int 0 to 1 &#40;1-e&#94;&#40;-x&#94;2&#41;&#41;&#40;1-e&#94;&#40;-y&#94;2&#41;&#41; dy dx - Wolfram|Alpha

I.e...

http://www.ballooncalculus.org/draw/double/four.png

To differentiate f(x.y) with respect to (e.g.) x, expand and differentiate term by term, treating y as constant.

Thanks for your help, I think I get it now!