Iterated Integral: int[-1,2] (int[0,e^(-x)] f(x, y) dy) dx

[warwick]

Can someone explain to me why we had to split this into two integrals? This is from my notes from class today.

 

[warwick]

Re: Iterated Integral

I think I just realized why. Up to whatever function value, the x value is from -1 to positive 2. After that, the x value is determined by - ln y. Is this correct?

 

[tkhunny]

Take a little stroll along the x- and y- axes. Look up and see what constitutes the ceiling. That would be your upper limit. (Well, in this case, the stroll should be along x = -1, not the y-axis.)

A view from the x-axis shows this region ALWAYS capped by y = e^(-x).

A view from the x = -1 shows the cap at y = e^(-x) (or its inverse) only part of the time.

 

[warwick]

All right.

I don't understand this change of order of integration.

I understand the 0 to 1 for dx but not 0 to x for dy.

Would it help to graph sin (x^2)?

Ok. It did help to see the graph. I now observe that y dx depends on the function, in this case, x. At first thought I thought the upper limit might be one, but the function looks like it doesn't reach one.