On changing the order of integration

[Emspak]

Homework Statement

This is a more general question, but I want to be sure I understand something here.

Take any function f(x,y) and say you're doing a double integral, like this:

$$\int^{\pi}_{\pi/2}\int^{sinx}_{\pi/2}f(x,y)dydx$$

So I want to reverse the order of integration. The x-coordinate interval is π/2 to π, and the y-interval is 0 to sinx.

So that means that x=arcsin y. and the interval in those terms starts at 1. Am I correct so far?

If I switch the order of integration I have to put the y interval on the outside. But since the y interval is in terms of x I have to change that. On this curve y is between 0 and 1. So I should make the outer integral from 1 to 0 since we were originally starting at π/2.

The inner integral is therefore π/2 to arcsin y.

And when you re-order the integral you should get

$$\int^{1}_{0}\int^{sin^{-1}}_{\pi/2}f(x,y)dxdy$$


But I know this is wrong and I am trying to make sure I understand why that is.

Any assistance is appreciated, as always.

 

[Mark44]

Homework Statement

This is a more general question, but I want to be sure I understand something here.

Take any function f(x,y) and say you're doing a double integral, like this:

$$\int^{\pi}_{\pi/2}\int^{sinx}_{\pi/2}f(x,y)dydx$$

So I want to reverse the order of integration. The x-coordinate interval is π/2 to π, and the y-interval is 0 to sinx.

Did you mean this integral?
$$\int^{\pi}_{\pi/2}\int^{sinx}_0f(x,y)dydx $$
This is what you described, below.

So that means that x=arcsin y. and the interval in those terms starts at 1. Am I correct so far?

If I switch the order of integration I have to put the y interval on the outside. But since the y interval is in terms of x I have to change that. On this curve y is between 0 and 1. So I should make the outer integral from 1 to 0 since we were originally starting at π/2.

The inner integral is therefore π/2 to arcsin y.

And when you re-order the integral you should get

$$\int^{1}_{0}\int^{sin^{-1}}_{\pi/2}f(x,y)dxdy$$

But I know this is wrong and I am trying to make sure I understand why that is.

For one thing, sin^-1 can't be one of the limits of integration. sin^-1(y) would be OK, though.

Assuming your first integral is as I corrected it, with the region of integration being bounded by the x-axis and y = sin(x), and between x = $\pi/2$ and $\pi$, the integral in reversed order would be
$$ \int_{y = 0}^1 \int_{x = \pi/2}^{sin^{-1}(y)}~f(x, y)~dx~dy$$

Any assistance is appreciated, as always.

 

[Emspak]

Yes, the correction you made was right (my inexpertise with itex entries. Oh well).

In any case, it seems i had my reasoning correct, even if my typing skills were off. So that's good to know.

Thanks Mark 44.

 

[SammyS]

Yes, the correction you made was right (my inexpertise with itex entries. Oh well).

In any case, it seems i had my reasoning correct, even if my typing skills were off. So that's good to know.

Thanks Mark 44.

I don't think all is right with the results.

If the original integral is $\displaystyle \ \int^{\pi}_{\pi/2}\int^{sinx}_0f(x,y)dydx \,,\$ then the region of integration satisfies the inequalies:

0 < y < sin(x) and π/2 < x < π .​

I.e. It's the region above the x-axis and below y = sin(x) that is to the right of x = π/2 and to the left of x = π .

The region of integration for the integral, $\displaystyle \ \int_{y = 0}^1 \int_{x = \pi/2}^{sin^{-1}(y)}~f(x, y)~dx~dy \$ is different because of the range of the arcsine function.
-π/2 ≤ arcsin(u) ≤ π/2



The upper limit of integration for the inner integral needs to be modified if the region of integration is to be to the right of x = π/2 .

 

[Emspak]

That wold mean that all I'd have to do tho, is subtract π from the arcsin function, no? So the range would be -π/2 to sin^-1(y)?