Changing Order of Triple Integration

[njo]

Homework Statement

\int_0^5 \int_0^2 \int_0^{4-y^2}\ \, dxdydx

Change order to dydxdz

Homework Equations

The Attempt at a Solution

I'm confused mainly because the limits are mostly numbers, not functions. I graphed the limits in @D and #d and this is what I got: \int_0^{4-y^2} \int_0^5 \int_0^2\ \, dydxdz

Not sure if its correct because I have a function on the outer integral, I can't check my work.

I'd appreciate any help. Thanks.

 

[LCKurtz]

Homework Statement

\int_0^5 \int_0^2 \int_0^{4-y^2}\ \, dxdydx

Change order to dydxdz

Homework Equations

The Attempt at a Solution

I'm confused mainly because the limits are mostly numbers, not functions. I graphed the limits in @D and #d and this is what I got: \int_0^{4-y^2} \int_0^5 \int_0^2\ \, dydxdz

Not sure if its correct because I have a function on the outer integral, I can't check my work.

I'd appreciate any help. Thanks.

It is clearly incorrect because you have variables in the outer integral limits. You need to start by sketching, or at least describing carefully, the volume being enclosed.

 

[SammyS]

Homework Statement

\int_0^5 \int_0^2 \int_0^{4-y^2}\ \, dxdydx

Change order to dydxdz

You obviously have a typo in the volume element there.

you have dx dy dx .

dx is repeated, no dz.

 

[HallsofIvy]

The limits on the "innermost" integration can be functions of the other two variables. The limits on the "middle" integration can be functions only of the one remaining variable and the limits of integration on the "outermost" integral must be constants. For example, integrating "dydzdx" we must have something like \int_a^b\int_{c(x)}^{d(x)}\int_{e(x,y)}^{f(x,y)} F(x,y,z) dydzdx.

With \int_0^2\int_0^{x^2}\int_0^{x+ y} F(x, y, z) dzdydx then x varies from 0 to 2, for each x, y varies from 0 to x^2, and, for each x and y, z varies from 0 to x+ y. If you were to graph the first two, it would be a "curved triangle" with one side the x-axis, one side the line x= 2, and the third side the curve y= x^2 which crosses x= 2 at (2, 4). If we only wanted to reverse dx and dy, to dzdxdy, note that, overall, y goes from 0 to 4 and that, for every y, x goes from 0 on the left to y= x^2, so x= \sqrt{y}. The integral, in that order, would be \int_0^4\int_0^\sqrt{y} F(x, y, z) dzdxdy.