Change of variables in double integral

In summary, to switch the bounds of a double integral, you need to rearrange the system of inequalities that describes the region of integration, which may involve splitting the region into multiple subregions and using inverse functions.
  • #1
yitriana
36
0
[tex]
\int_{c_1}^{c_2} \int_{g_1 (x)}^{g_2 (x)} f(x,y) dy dx[/tex]

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:

[tex]
\int_{0}^{4} \int_{2}^{\sqrt{y}} e^{x^3} dx dy[/tex],

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

[tex]
\int_{0}^{2} \int_{0}^{x^2} e^{x^3} dy dx[/tex]

and make it possible to integrate.

==

So, how would we rewrite something like

[tex]
\int_{0}^{2} \int_{\sin{y}}^{(y-1)(y-2)(y-3)} e^{x^3} dx dy[/tex]

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?
 
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  • #2
yitriana said:
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:
[tex]c_1 \leq x \leq c_2[/tex]
[tex]g_1(x) \leq y \leq g_2(x)[/tex]
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:
[tex]\int_{c_1}^{c_2} \int_{g_1(x)}^{g_2(x)} f(x, y) \, dy \, dx = \int_S f[/tex]

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
[tex]-1 \leq x \leq 1[/tex]
[tex]2x^2 - 1 \leq y \leq x^2[/tex]​
(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: [itex]y \leq 0[/itex]
Region II: [itex]y > 0[/itex] and [itex]x \leq 0[/itex]
Region III: [itex]y > 0[/itex] and [itex]x > 0[/itex]

(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
[tex]y \leq 0[/tex]
[tex]-1 \leq x \leq 1[/tex]
[tex]2x^2 - 1 \leq y \leq x^2[/tex]​

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.
 

1. What is a change of variables in a double integral?

A change of variables in a double integral involves transforming the given variables of integration into new variables, which simplifies the integral and makes it easier to solve. This is a common technique used in multivariable calculus to solve complex integrals.

2. Why is a change of variables useful in double integrals?

A change of variables can make a double integral easier to solve by reducing the complexity of the integrand. It also allows us to change the region of integration, making it more convenient to find the solution.

3. How do you determine the new limits of integration after a change of variables?

To determine the new limits of integration, we need to consider how the original variables are related to the new variables. We can use this relationship to transform the original limits into the new variables, giving us the new limits of integration.

4. What are some common substitutions used in a change of variables for double integrals?

Some common substitutions used in a change of variables for double integrals include polar coordinates, cylindrical coordinates, and spherical coordinates. These substitutions are often used to simplify the integrand and the region of integration.

5. Can a change of variables be used in any type of double integral?

No, a change of variables can only be applied to certain types of double integrals, such as those with rectangular or circular regions of integration. It may not be useful or possible to use a change of variables in more complex regions of integration, such as those with irregular shapes or holes.

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