Solving Nested Double Integration Problems

[ApeXaviour]

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:

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

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:

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

Cheers
-Dec

 

[arildno]

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.

 

[ApeXaviour]

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) ?

 

[arildno]

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.

 

[ApeXaviour]

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.

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..

 

[amcavoy]

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.

 

[ApeXaviour]

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..

 

[amcavoy]

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...

 

[ApeXaviour]

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.