Question

how do you solve double integrals?

Answer 1

integrate, twice

Answer 2

You integrate as you would normally any integral with respect to each variable. For example, \[\int\limits \int\limits xdy dx\] means take the integral of y.
Treat x as a constant. So you would have \[\int\limits xy dx\]
Now, you can take the integral of x.

However, double and triple integrals is usually dealing with regions and hence, they will usually be definite integrals or integrals over general regions, meaning functions and similarly, polar coordinate integrals.

Answer 3

According to Fubini's theorem:

If f(x,y) is continuous on R = [a,b] x [c,d] then: \[\int\limits \int\limits_{R}^{ }f(x,y)dA = \int\limits_{a}^{b} \int\limits_{c}^{d}f(x,y)dydx = \int\limits_{a}^{b} \int\limits_{c}^{d} f(x,y)dxdy\]

Answer 4

for something like definite integrals, after finding dx wouldn't you have two parts (the first limit minus the second limit)? and then you have to find the dy integral of that?? is it like that for all problems??? thank you!!

Answer 5

Let me give you an example

Answer 6

ok! i am so just so confused after doing the first integral T.T

Answer 7

\[\int\limits \int\limits_{R}^{ }6xy^2dA\] \[R = [2,4] \times [1,2]\] is really: \[\int\limits_{2}^{4}\int\limits_{1}^{2}6xy^2dydx = \int\limits_{1}^{2}\int\limits_{2}^{4}6xy^2dxdy\]
according to fubini's theorem, you can switch whichever you want to integrate first and you will still get the same answer! Notice that i switched which variable i would integrate first! It can either be dy or dx first or dx dy...
If I followed by integrating with respect to y first, then the solution would be: \[\int\limits_{2}^{4}\left[ 2xy^3 \right]\] evaluating from 2 to 1
and then I would do the second integral with respect to x: \[\int\limits_{2}^{4}16x-2xdx = \int\limits_{2}^{4}14xdx\]

Answer 8

ohhh so before you do the second integral you have to actually evaluate the value of the first integral so by the time you solve the second integral its down to only one variable??

Answer 9

Yep.

Answer 10

ok thank you very much!! :)