- #1

- 655

- 0

I am able to manipulate and use double integrals, but I am having a bit of mental block when trying to visual how they actually work.

First, would you agree that a double integral is simply summing a function over a region by taking lots of tiny squares (or rectangles?) of sides dx and dy => dA = dxdy.

So let's take http://gyazo.com/39f756b268c4e1cf4d8e09b7213e58ed

So we want to integral the function over the triangular region. This is where the trouble comes in. As I said I can do it, but I cannot really visualize exactly how the limits work

So the internal integral is integrating with respect to x. To me the area we need to compute is

http://gyazo.com/ed588350139dd383b783542ce8e3ab11

So the limits go from 0 to a NOT x=y to a. All the limits tell you is where to begin integrating from. y is simply the height of the function at each strip (dx - shown in red), not an actual limit

Similarly, if we had y as the internal integral

http://gyazo.com/eafcfb2b70f96c9c6b517167ceea29fb

The limits would be (y = x to y=a) but y=x tells you how wide your strip (dy - shown in blue is). If you look where the y strips start from , they go from y = 0 to y = a,

Can someone please explain to me where I have gone wrong in my reasoning/provide a better visualization/interpretation

Thanks

Thomas