Solving Nested Double Integration Problems

  • Thread starter ApeXaviour
  • Start date
  • Tags
    Integration
In summary, the person is looking for help with an integration question and hit a snag. They went through various resources but still couldn't remember how to do it. They eventually figured it out by doing the x-integration first and found that the same problem occurred when they tried to do the y-integration.
  • #1
ApeXaviour
35
0
Right, I was looking over some past exam papers from these science-maths scholarship exams I was looking to do. It's all revision for me but it's been a while. Anyway, skimming over the double integration questions. Now if I recall correctly it's a fairly simple process, integrate the inner bit, then the outer et voila right? Ok you got a bit more with oddly shaped areas (type I and II etc) and whatnot but it's still pretty straightforward.

I hit a bump, embarrassingly just with the mechanics of integrating itself, I just couldn't damn remember what to do when the question came up like this.

To elaborate, I'd come across this question:
1.GIF


Now, if I'm doing this correctly I initially ignore the outer nested bits and concentrate on this:
2.GIF


Wtf do I do with that? I put it through mathematica and it comes up with some "null" argument jive.
Skimmed numerous times through all relevant sections of my calculus book (by Anton) to no avail..

Looked at the previous year's question and sure enough there was one just like it:
3.GIF


With no such example in my obviously inferior recommended text I'm left to ask for help.

Cheers
-Dec
 
Physics news on Phys.org
  • #2
Hint:
Change the description of your region so that "y" runs from 0 to 2, and x runs from 0 to 1/2y.

EDIT:
I ended up with something like 13/18.
 
Last edited:
  • #3
I'm at a loss, why would I do that? The limits are 2 to 2x for y and 0 to 1 for x, where did you get those other numbers from?
And I don't understand how changing the limits would help at all, it's before that. I mean how does one integrate sqrt(y^3 + 1) ?
 
  • #4
ApeXaviour said:
I'm at a loss, why would I do that? The limits are 2 to 2x for y and 0 to 1 for x, where did you get those other numbers from?
DRAW the region in the xy-plane!
See that your region could equally well be described by my limits.
And I don't understand how changing the limits would help at all, it's before that. I mean how does one integrate sqrt(y^3 + 1) ?
Do the x-integration first, and you'll see that all your problems are solved.
 
Last edited:
  • #5
arildno said:
DRAW the region in the xy-plane!
See that your region could equally well be described by my limits.
Ah, my apologies, I see where you get this now. Thank you. It has been over a year since I've studied mathematics.

arildno said:
Do the x-integration first, and you'll see that all your problems are solved.
Okay I did this with the new limits but I still came back to the same problem. After doing the x-integration I get:
1/4 *y^2 * sqrt(y^3 + 1). Which I can't integrate for y, I tried substitution and tabular integration. Looked over the previous integration and couldn't see any slip.

Thanks for your patience so far by the way..
 
  • #6
ApeXaviour said:
Ah, my apologies, I see where you get this now. Thank you. It has been over a year since I've studied mathematics.


Okay I did this with the new limits but I still came back to the same problem. After doing the x-integration I get:
1/4 *y^2 * sqrt(y^3 + 1). Which I can't integrate for y, I tried substitution and tabular integration. Looked over the previous integration and couldn't see any slip.

Thanks for your patience so far by the way..

Try u=y3+1.
 
  • #7
apmcavoy said:
Try u=y3+1.
OH MY GOD.. I'm so slow today. It's obvious. Thanks

This is what I get for not studying anything the last 5 months. Brainrot.. :redface:
 
  • #8
Looked at the previous year's question and sure enough there was one just like it: http://www.maths.tcd.ie/~cockburd/3.GIF

Was this the actual question? It seems like it should be a double integral...
 
  • #9
apmcavoy said:
Was this the actual question? It seems like it should be a double integral...
The actual question is to evaluate this:
http://www.maths.tcd.ie/~cockburd/4.GIF

Which I think is a missprint (I'd heard there was one on this paper) as otherwise I found it impossible to do. I substituted y/2 for x/2 and it worked out like the previous one.
 

1. What is the process for solving nested double integration problems?

The process for solving nested double integration problems involves breaking down the problem into smaller integrals, then using the power rule, product rule, and chain rule to integrate each term. The inner integral is integrated first, using the appropriate rules, and then the outer integral is integrated using the same rules. This process may need to be repeated multiple times depending on the complexity of the problem.

2. How do I know which variable to integrate first when solving nested double integration problems?

The variable to integrate first is typically the one that has the fewest number of constants attached to it. This variable is known as the inner integral. However, there may be situations where it is more efficient to integrate the outer integral first. It is important to carefully examine the problem and decide which variable to integrate first based on the constants and functions involved.

3. What are some common mistakes to avoid when solving nested double integration problems?

One common mistake is forgetting to integrate the inner integral before moving on to the outer integral. Another mistake is incorrectly applying the power rule, product rule, or chain rule. It is also important to carefully track constants and variables throughout the integration process to avoid errors. It is recommended to always double check the final answer by differentiating it to ensure it is correct.

4. How can I simplify nested double integration problems?

Simplifying nested double integration problems can be done by using integration by parts, trigonometric substitution, or other integration techniques. It may also be helpful to identify and use symmetry in the problem to simplify the integrals. Additionally, using algebraic techniques to simplify the integrand before integrating can make the process easier.

5. Are there any tips or tricks for solving nested double integration problems?

One helpful tip is to use a table or chart to keep track of the integrals and their solutions as you work through the problem. This can help you avoid mistakes and make the process more organized. It is also important to carefully check the limits of integration and ensure they are correctly applied to each integral. Additionally, practice and familiarity with integration techniques can help make solving nested double integration problems easier over time.

Similar threads

Pendulum Question (Matlab Coursework)
    • Introductory Physics Homework Help
Replies
4
Views
1K
Double Integral via Appropriate Change of Variables
    • Calculus and Beyond Homework Help
Replies
3
Views
925
Faraday's law with Calc 3 integration help
    • Introductory Physics Homework Help
Replies
1
Views
1K
Solving a Puzzling Physics Problem: ##15 \pm 3\sqrt{15}##
    • Introductory Physics Homework Help
Replies
17
Views
1K
Finding the Fourier cosine series for ##f(x)=x^2##
    • Calculus and Beyond Homework Help
Replies
4
Views
617
A Help needed with derivation: solving a complex double integral
    • General Math
Replies
1
Views
742
Operators On Multivariable Wave Functions
    • Advanced Physics Homework Help
Replies
8
Views
364
Magnetic flux with magnetic field changing direction
    • Introductory Physics Homework Help
Replies
7
Views
1K
Previous university exam problem on refraction (that I think is wrong)
    • Introductory Physics Homework Help
Replies
14
Views
2K
I In which sense(s) do square integrable functions go to zero at infinity?
    • Topology and Analysis
Replies
2
Views
151

Hot Threads

Recent Insights

Back
Top