integrating implicit functions

http://en.wikipedia.org/wiki/Integration_by_parts ??

That would be integrating f(x) * g(x) ?

I dont think you can, or maybe you can but it would be useless.

When you differentiate an implicit function, you are differentiating each term with respect to a rate of change to another variable.

When integrating an explicit function, the function itself is a rate of change of its area, so when you have an implicit function and you integrate it you might mess things up

Though im not sure

hmm, ok, thanks for your help

shortref, wouldn't it be useful if you were given a derived function in the form f(xy) and you wanted to find the original function?

If you were given a derived function in the form f(xy) it would just be a differential equation.

Say $x^2 + y^2 = 1$

differentiating this with respect to x

$2x + 2y = 0$

If you wanted to integrate this, you wouldnt need any magical 'implicit integration' but you just need to solve this differential equation.

can you do that for something like, say

$\frac{dy}{dx} = xsin(x+y)$

i was looking at differentiating implicitly ... which got me thinking... how would you integrate an implicit function?

obviously you can do certain ones by recognition but is there a method to do stuff like this?

google didn't shed much light...

any help is much appreciated

then

http://en.wikipedia.org/wiki/Multiple_integral

Do you mean functions of the sort F(x,y) = 0?
Which ones can you do by recognition?

can you do that for something like, say

$\frac{dy}{dx} = xsin(x+y)$

The volume of a sphere one you could probably do by recognition:

$\mathrm{Volume} = \int_0^{2 \pi }\, d \phi \int_0^{ \pi } \sin \theta\, d \theta \int_0^R \rho^2\, d \rho = 2 \pi \int_0^{ \pi } \sin \theta \frac{R^3}{3 }\, d \theta = \frac{2}{3 } \pi R^3 [- \cos \theta]_0^{ \pi } = \frac{4}{3 } \pi R^3.$

But actually I dont think there are any multivariable functions that you can integrate by recognition

I thought the OP was asking: can you do $\int f(x,y) \ dx$? In which case the answer is a straightforward no.

Do you mean functions of the sort F(x,y) = 0?
Which ones can you do by recognition?

Whats wrong with this? Apart from that fact that its definite?

i was thinking about stuff like



which you could recognise as the gradient function of

$x^2 + y^2 = c$

although i phrased it in a pretty crap way.

so it's a big no to integrating stuff given implicitly?

i was thinking about stuff like



which you could recognise as the gradient function of

$x^2 + y^2 = c$

although i phrased it in a pretty crap way.

so it's a big no to integrating stuff given implicitly?