Triple Integral - Change the order of integration

[sirhc1]

Homework Statement

$\int^{1}_{0}\int^{x^2}_{0}\int^{y}_{0} f(x,y,z) dz dy dx$

Find 5 equivalent iterated integrals.

Homework Equations

$0 ≤ z ≤ y$

$0 ≤ y ≤ x^2$

$0 ≤ x ≤ 1$

The Attempt at a Solution

1) $\int^{1}_{0}\int^{√y}_{0}\int^{x^2}_{0} f(x,y,z) dz dx dy$

I will try dz dy dx first.

Because y = x^2, so $0 ≤ z ≤ x^2$

Because y = x^2, so $0 ≤ x ≤ √y$

And by the same logic, $0 ≤ y ≤ 1$

When I integrate for f(x,y,z) = 1, the correct answer is 1/10. I do not get the same answer with my solution. Help! Is it possible to solve this without graphing it? Or is it necessary to get the correct answer?

 

[sirhc1]

Just noticed the SAME question was asked 3 days ago.

https://www.physicsforums.com/showthread.php?t=554227&highlight=change+the+order+of+integration

@HallsofIvy, I came up with the same solution as you for dxdydz. However, when I integrate for f(x,y,z) = 1, the answer is incorrect. I do not understand what is the flaw in my and your method?

 

[HallsofIvy]

Homework Statement

$\int^{1}_{0}\int^{x^2}_{0}\int^{y}_{0} f(x,y,z) dz dy dx$

Find 5 equivalent iterated integrals.

Homework Equations

$0 ≤ z ≤ y$

$0 ≤ y ≤ x^2$

$0 ≤ x ≤ 1$

The Attempt at a Solution

1) $\int^{1}_{0}\int^{√y}_{0}\int^{x^2}_{0} f(x,y,z) dz dx dy$

I will try dz dy dx first.

Because y = x^2, so $0 ≤ z ≤ x^2$

Because y = x^2, so $0 ≤ x ≤ √y$

Here is an error. For each x, y goes from 0 to $x^2$. If you graph that region in an xy-plane, it is below and to the right of the graph of $y= x^2$. That means that, for each y, x goes from $\sqrt{y}$ up to 1. The y-integral is $\int_{x^2}^1 dy$

Looking at this, I realize now that I made that mistake in the previous post. I have edited it.

And by the same logic, $0 ≤ y ≤ 1$

When I integrate for f(x,y,z) = 1, the correct answer is 1/10. I do not get the same answer with my solution. Help! Is it possible to solve this without graphing it? Or is it necessary to get the correct answer?