Change of variables in double integral

[yitriana]

$$\int_{c_1}^{c_2} \int_{g_1 (x)}^{g_2 (x)} f(x,y) dy dx$$

If f(x,y) is function such that it is not easily integrable, if we wanted to switch the bounds of integration so that h₁(y) = g₁(x) , same for g₂(x),
what would be the general way to rewrite the bounds? Would it involve inverse functions?

Let me clarify with a specific (easy) example:

$$\int_{0}^{4} \int_{2}^{\sqrt{y}} e^{x^3} dx dy$$,

rewriting sqrt(y) = x as y = x², and finding intersection points would enable us to rewrite as,

$$\int_{0}^{2} \int_{0}^{x^2} e^{x^3} dy dx$$

and make it possible to integrate.

==

So, how would we rewrite something like

$$\int_{0}^{2} \int_{\sin{y}}^{(y-1)(y-2)(y-3)} e^{x^3} dx dy$$

in solvable terms?

Would we have to find the inverse of sin{y} and the other function as a function of x and make those the bounds for dy?

 

[Hurkyl]

what would be the general way to rewrite the bounds? Would it involve inverse functions?

Geometry helps. Your bounds describe a two-dimensional region S in the Euclidean plane. Explicitly, S is the set of all solutions to the system of inequations:
$$c_1 \leq x \leq c_2$$
$$g_1(x) \leq y \leq g_2(x)$$
and the main theorem (Fubini) you're invoking is that if f is a sufficiently nice function, then this iterated (one-dimensional) integral is equal to the (two-dimensional) integral over the region:
$$\int_{c_1}^{c_2} \int_{g_1(x)}^{g_2(x)} f(x, y) \, dy \, dx = \int_S f$$

If you want to switch the order of integration, then you need to rearrange the above system of equations. Quite frequently, this requires splitting the region S into several subregions. For example, suppose S was given by

$$-1 \leq x \leq 1$$
$$2x^2 - 1 \leq y \leq x^2$$​

(Sketch this region on a sheet of paper)

(No seriously, go sketch it before you continue reading)

I think (hope) it's clear that you should separate S into three regions:

Region I: $y \leq 0$
Region II: $y > 0$ and $x \leq 0$
Region III: $y > 0$ and $x > 0$

(These three regions cover the entire plane. This makes sure that I don't accidentally "lose" or "overcount" any points -- every point in S is guaranteed to be in exactly one of these three regions. It would have been okay to let them overlap along a line, though)

In Region I, my system of equations is

$$y \leq 0$$
$$-1 \leq x \leq 1$$
$$2x^2 - 1 \leq y \leq x^2$$​

It should be clear how to express this region as a system of inequalities that let's you write down an iterated integral with the variables reversed. You can grind it out through purely algebraic manipulation if you wanted to (including discovering the way to split it into three regions), but it's not an exercise I would recommend except as algebra practice.