make the interface to integrate() like the (new consistent) interface to diff()

[jason]

It would be nice if the following worked, if f was a function:

sage: integrate(f, x)

sage: # Double indefinite integral wrt x
sage: integrate(f, x, x)

sage: # limits and a double integral: x is the inner integral, y is the outer integral (note that this order is backwards from Mma...they think of nested integrals as int dx int dy function, so the first parameter is the outer integral in Mma.)
sage: integrate(f, (x, 0, 1), (y, 1, 2))
sage: integrate(f, (x, 0, y), (y, 1, 2))

sage: # Double integral, x is the inner integral, y is the outer integral
sage: integrate(f, x, y)

[jwmerrill]

See also #1221. It would be nice if the interface here was also consistent with the interface to plot, at least when a range is specified, since both take a function, a variable, and a range.

[jwmerrill]

What do you propose to do about the following case:

# Current behavior, definite integral with x and y as bounds
sage: var('x,y')
sage: integrate(x^2,x,x,y)
y^3/3 - x^3/3

# Triple Integral, twice with respect to x, once with respect to y
sage: integrate(x^2,x,x,y)
x^4*y/12

# i.e. same as
sage: (x^2).integrate(x).integrate(x).integrate(y)

[jason]

As noted in #1221, I would vote with the consensus from the devel list that the syntax integrate(f, x, a, b) be deprecated in favor of integrate(f, (x, a, b)). However, you're right that this would introduce a backwards-incompatible change.

[nbruin]

I would say that deprecating integral(f(x),x,a,b) for a definite integral will be too large a backward incompatible change to pull of.

The gain for multiple integrals is also a little tricky: The proposed order for integral(f(x,y),x,y) where x is inner and y is outer is definitely consistent with mathematical notation int int f(x,y) dx dy but it's opposite to python's iteration order [ (i,j) for i in range(3) for j in range(3) ], where j iterates in the inner loop.

Also, the proposed notation is for multiple integrals, which in mathematics are usually written as multiple (nested) integrals anyway. Single integral notation is usually reserved for higher dimensional integrals (Lebesgue, usually), in which case no particular "integration order" is implies, and indeed the domain of integration might not allow for a direct translation into nested integrals.

The fact that integral has to both accommodate definite integrals and taking antiderivatives breaks the complete symmetry with the diff interface. It would be nice if the interface were more symmetrical, but I think there are good reasons why it's not. If a complete converse to diff() is required, perhaps we need antiderivative(), which doesn't need to accommodate definite integrals (although of course no-one would find its existence).

[kcrisman]

I think one big meta-motivation behind this is that some competitor programs may be more consistent or at least allow multiple integrals this way? Certainly it seems much more cumbersome in Sage - plus, you have to do them in the "right order" to get nice answers sometimes, I would guess, just like in calculus textbooks :)