Oxbridge Physics Prelims Revision

seeing that I have a maths exam tomorrow, I should probably do some revision (although I feel pretty hopeless at trying to learn stuff which I cant really do anyway...you could say I was lucky I had questions I could manage on monday but still). Just a quick question:

Find the derivative with respect to a of:

$\int_{0}^{a^2} \frac{\sin ax}{x} dx$

I know you can flip the order of integration/differentiation hence write:

$\int \frac{\partial}{\partial a}\frac{\sin ax}{x} dx = \int \cos ax dx$

but how do you change the limits in this case? (since the limits are themselves functions of a)

cheers!

I get sin(a^3)/a

But am in a rush so will have to explain later

Try setting up the integral the integral I(a,x) = Int[t:0_>x] sin(at)/t dt

Work out I's partial derivatives wrt a and x

Then use the chain rule to work out the derivative of I(a,a^2)

*gasp* RichE must have got something wrong!!! That's exactly the answer I handed in back ages ago when I did this question and it is wrong. Because I said you do the integral on the cos ax between a^2 and 0, and you'll get your 1/a *sin a^3. But that's definitely wrong. I've got a note scribbled next to it saying "You must change the limits -> They are functions of a"....but I cant remember how to apply that!

And sorry I dont have the answer written down i'm afraid, but that answer you just gave is definitely wrong! (i think)

is this simon or chloé?

Well I certainly didn't get my wrong answer that way! If you'd read my hint you would have realised this. Though as ever your response is just dripping with gratitude at the time I spend on these questions.

Checking my working now -

I(a,x) = Int[t:0->x] sin(at)/t dt

I_2(a,x) = sin(at)/t

I_1(a,x) = Int[t:0->x] pd/pda (sin(at)/t) dt = Int[t:0->x] cos(at) dt

= sin(ax)/a

[this is where my error came from previously, I'd got this bit as -sin(ax)/a]

You are interested in F(a) = I(a,a^2)

So F'(a) = I_1(a,a^2) + 2a I_2(a,a^2) =

sin(a^3)/a + 2a sin(a^3)/a^2 = 3sin(a^3)/a

Which I hope is ok - I have never made any claims of infallibility - but frankly with your attitude to my posts I don't know why I f*cking bother

It was me (Chloé). Simon and two of the other Lincoln Physicists made me drink a bottle of cava and 2 pimms very quickly on an empty stomach straight after I'd finished. He then dragged me back to college on the bus (apologies to anyone who was on it at the same time as me) and left me in my room for an hour while he went to sing with OOTB. I was then sick and thankfully that sobered me up somewhat (I came to the computer when my friend rang my internal number and discovered Simon had been using it and then somehow I found TSR!). Have spent 3hours this afternoon punting and we're off to Pizza Hut soon to eat and am now completely sober (thank god as meeting parents at 7pm for theatre!!!).

Well done to everyone who's finished and good luck to everyone (Willa & Camford) who still have them to go

Yeah you didnt look too good. I was gonna say hi and chat and so on but you didnt look at all well so I thought I would leave it. Hope you are feeling better/ less drunk now

Howd exam go? I think It was really good. Im pretty much sure I nailed it. I finished with 10 mins to spare. I love quantum! I just wish theyd been like that last week.

It wasn't amazing for me but then I'm not really suprised as I could never do the topic. I think I possibly managed to get the 20marks needed to pass. Now all we can do is sit and wait for our results...well actually go and have lots of fun! YAY!

As it turns out it wasnt the method that was wrong just the working, so I'll have a crack your working.

I think we all know on here how much the methods adopted at cambridge and oxford differ....whenever i have posted my reasoning on here to a question, it has often greatly differed from that of oxford physicists.

I need to understand WHY you are doing something on each line, and I have found your answers to physics problems a tad to self-consistent for me to handle (and hence why I challenge the reasoning, cos I dont understand it).
Take that whole relativity stuff from a week back. From your reasoning all I could deduce was a sort of circular consistency: The lorentz transformation is correct, hence you have this change in time when measuring something, hence the lorentz transformation is correct.

seeing that I have a maths exam tomorrow, I should probably do some revision (although I feel pretty hopeless at trying to learn stuff which I cant really do anyway...you could say I was lucky I had questions I could manage on monday but still). Just a quick question:

Find the derivative with respect to a of:

$\int_{0}^{a^2} \frac{\sin ax}{x} dx$

I know you can flip the order of integration/differentiation hence write:

$\int \frac{\partial}{\partial a}\frac{\sin ax}{x} dx = \int \cos ax dx$

but how do you change the limits in this case? (since the limits are themselves functions of a)

cheers!

i personally would just do integration by parts. take u=sinax and dv=1/x, then you get du=acosax and v=lnx, then you do the uv - ∫v.du and thats sinax.lnx - a∫lnx.cosax.dx. Then you do integration by parts again, but let u=lnx this time and it probably works out eventually. If you do it enough times (maybe four?) you end up with taking away the original integral which you can just take over to the other side. Or something.

Can I just ask, was that this term's work?

edit: does integration by parts even work for that?

*resolves not to answer any stupid integration questions tomorrow*

i personally would just do integration by parts. take u=sinax and dv=1/x, then you get du=acosax and v=lnx, then you do the uv - ∫v.du and thats sinax.lnx - a∫lnx.cosax.dx. Then you do integration by parts again, but let u=lnx this time and it probably works out eventually. If you do it enough times (maybe four?) you end up with taking away the original integral which you can just take over to the other side. Or something.

You are interested in F(a) = I(a,a^2)

So F'(a) = I_1(a,a^2) + 2a I_2(a,a^2) =

RichE I dont get this bit in your working:



How are you relating I to I_1 and I_2?
cheers