Changing the order of a triple integration

Click For Summary
SUMMARY

The discussion focuses on changing the order of integration for the definite integral \(\int_0^{1}\int_{\sqrt{x}}^{1}\int_{0}^{1-y}f(x,y,z)dzdydx\) to the order dydxdz. The original limits of integration are defined as z ranging from 0 to 1 - y, y from sqrt(x) to 1, and x from 0 to 1. To determine the new limits for y, participants emphasize the importance of sketching the region of integration, which aids in visualizing and establishing the correct boundaries for the new order.

PREREQUISITES
  • Understanding of triple integrals in calculus
  • Familiarity with changing the order of integration
  • Ability to sketch regions in three-dimensional space
  • Knowledge of definite integrals and their limits
NEXT STEPS
  • Study techniques for changing the order of integration in triple integrals
  • Learn how to sketch regions of integration in three dimensions
  • Explore examples of definite integrals with varying limits
  • Review the concept of iterated integrals and their applications
USEFUL FOR

Students and educators in calculus, mathematicians working with integrals, and anyone looking to deepen their understanding of multivariable integration techniques.

paraboloid
Messages
17
Reaction score
0
I'm given this definite integral:
[itex]\int_0^{1}\int_{\sqrt{x}}^{1}\int_{0}^{1-y}f(x,y,z)dzdydx[/itex]

I need to change the order to dydxdz, but I'm stuck trying to get the limits of integration wrt y.

24on636.png


[itex]\int_0^{1}\int_{x^2}^{0}\int_{}^{}f(x,y,z)dydzdx[/itex]

How do I find the limits of y?
 
Physics news on Phys.org
Just look at the original iterated integral to see the region over integration occurs.

z = 0 to z = 1 - y
y = sqrt(x) to y = 1
x = 0 to x = 1.

Sketch that region and it will help you determine the limits of integration for a different order of integration.
 

Similar threads

  • · Replies 13 ·
Replies
13
Views
3K
  • · Replies 2 ·
Replies
2
Views
2K
Replies
4
Views
3K
  • · Replies 10 ·
Replies
10
Views
2K
  • · Replies 19 ·
Replies
19
Views
3K
Replies
2
Views
2K
Replies
2
Views
1K
Replies
9
Views
2K
  • · Replies 9 ·
Replies
9
Views
2K
  • · Replies 2 ·
Replies
2
Views
1K