Triple Integral - Change the order of integration

  • Thread starter sirhc1
  • Start date
  • #1
4
0

Homework Statement



[itex]\int^{1}_{0}\int^{x^2}_{0}\int^{y}_{0} f(x,y,z) dz dy dx[/itex]

Find 5 equivalent iterated integrals.

Homework Equations



[itex]0 ≤ z ≤ y[/itex]

[itex]0 ≤ y ≤ x^2[/itex]

[itex]0 ≤ x ≤ 1[/itex]

The Attempt at a Solution



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

I will try dz dy dx first.

Because y = x^2, so [itex]0 ≤ z ≤ x^2[/itex]

Because y = x^2, so [itex]0 ≤ x ≤ √y[/itex]

And by the same logic, [itex]0 ≤ y ≤ 1[/itex]

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?
 

Answers and Replies

  • #3
HallsofIvy
Science Advisor
Homework Helper
41,847
966

Homework Statement



[itex]\int^{1}_{0}\int^{x^2}_{0}\int^{y}_{0} f(x,y,z) dz dy dx[/itex]

Find 5 equivalent iterated integrals.

Homework Equations



[itex]0 ≤ z ≤ y[/itex]

[itex]0 ≤ y ≤ x^2[/itex]

[itex]0 ≤ x ≤ 1[/itex]

The Attempt at a Solution



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

I will try dz dy dx first.

Because y = x^2, so [itex]0 ≤ z ≤ x^2[/itex]

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

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

And by the same logic, [itex]0 ≤ y ≤ 1[/itex]

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?
 
Last edited by a moderator:

Related Threads on Triple Integral - Change the order of integration

Replies
12
Views
4K
Replies
5
Views
2K
Replies
17
Views
1K
Replies
6
Views
14K
Replies
1
Views
3K
Replies
1
Views
1K
Replies
3
Views
2K
  • Last Post
Replies
3
Views
897
Replies
1
Views
2K
Replies
13
Views
1K
Top