Examples of using Double integral in English and their translations into Polish
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What does this double integral represent?
So it should really just be the double integral.
In double integral problems the hardest part is figuring out the boundaries.
So that's why it's a double integral.
It's a double integral here because we want to add up all of the d sigmas in 2 directions.
You could have just done a double integral.
This is the same thing as the double integral over the region over which our parameters are defined.
And then the video after that we will actually evaluate this double integral.
It should be equal to the double integral over the surface.
It might be obvious to you if you have recently done some double integrals.
This calculator calculates definite double integrals of real functions with two real variables.
to multivariable calculus double integrals.
In all of the double integrals we have done so far,
And we really need to re-express it in terms of a double integral in the domain of the parameters.
with respect to z, we ended up with a double integral.
You hopefully have a little intuition now on what a double integral is or how we go about figuring out the volume under a surface.
then evaluate this double integral.
Let me write that in that same-- it will be the double integral over our region, which is really just the same thing as our surface, of the curl of f dot n.
But if you just saw an integral like this, this is the inside of a double integral. And it is.
The double integral over the region of, I will write this one first because it's positive,
The line integral of the closed loop of F dot dr is equal to the double integral of this expression. And it's something.
So we're going to do a double integral, a two-dimensional integral,
how it relates to a double integral, and then later in the next video we could do something slightly more complicated.
And this double integral is the exact integral we would have done in the previous videos on the double integral,
we were able to connect its line integral to the double integral over region-- oh,
When we say that we're just taking the double integral, first of all, that tells us we're doing it in two dimensions,
assuming it has these partial derivatives, to the region, to a double integral of the region.
And then even once we evaluated, once we integrated with respect to z, we ended up with a double integral, which is exactly what you would have done in the last several videos when we just learned the volume under a surface.
And then that whole thing-- I'm not going to take the absolute value because I need a vector right over here-- times differentials of the parameters-- d theta dr. And we can swap these two things around depending on what will make our eventual double integral easier.
So that's plus, I will leave it up there, maybe I will do it in the yellow, plus the double integral over the same region of the partial of Q with respect to x. da,