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?