ATAR Notes: Forum

VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Mathematical Methods CAS => Topic started by: /0 on August 22, 2008, 08:41:46 pm

Title: Properties of integrals
Post by: /0 on August 22, 2008, 08:41:46 pm: Find $\int_0^3 f(3x)\,dx$ if $\int_0^9 f(x) \,dx = 5$ .

Find $\int_0^1 f(3x+1)\,dx$ if $\int_1^4 f(x) \,dx =5$

I don't remember these properties of integrals being taught :/ I can kinda see how this works through dilations etc. but I'm not sure about it.

Thanks
Title: Re: Properties of integrals
Post by: orsel on August 22, 2008, 08:48:34 pm: substitution:
u=3x, du/dx=3 etc
Title: Re: Properties of integrals
Post by: /0 on August 22, 2008, 08:50:41 pm: Oh yeah, thanks. I guess that works.

Substitution isn't in the methods course though ?
Title: Re: Properties of integrals
Post by: Mao on August 22, 2008, 09:06:55 pm: substitution isnt in the methods course, but these particular problems can be thought of graphically as transformations of an area (e.g. in the first case, dilation)
Title: Re: Properties of integrals
Post by: orsel on August 23, 2008, 10:33:10 am: oh yeah I forgot that lol, sorry, just the first thing that came to mind

but the examiners do accept any method that is mathematically correct, even if its outside the course.
Title: Re: Properties of integrals
Post by: shinny on August 23, 2008, 01:36:37 pm: is what orsel said true? because i don't think it is. from what i know, you're not allowed to double diff in methods even if it is 'mathematically correct'. it'd save the trouble of sign tables though <_<
Title: Re: Properties of integrals
Post by: orsel on August 23, 2008, 01:49:31 pm: It is true, if you believe me. My teacher was an examiner for methods and spesh up to last year, so I trust her on this.

Edit: not for both methods and spesh at same time, I think, but in different years.
Title: Re: Properties of integrals
Post by: ed_saifa on August 23, 2008, 01:55:33 pm: Quote from: orsel on August 23, 2008, 01:49:31 pm
It is true, if you believe me. My teacher was an examiner for methods and spesh up to last year, so I trust her on this.
It is true. It just needs to be mathematically correct.
Title: Re: Properties of integrals
Post by: Glockmeister on August 23, 2008, 02:45:56 pm: My teacher seems to say the same thing... then my friend said this

"You know too much. Fail!"
Title: Re: Properties of integrals
Post by: /0 on August 23, 2008, 09:31:55 pm: Oh another question, two questions...

1. If you have a graph of growth rate of penguins against time and you integrate under the curve, would you say
$Area = \int_a^b n'(t) \,dt$ penguins
Or would you just say sq. units?

2. Show that the curves with equations $y=x^n$ and $x=y^n$ intersect at (1,1) for $n=1,2,3,...$ .
Title: Re: Properties of integrals
Post by: Mao on August 23, 2008, 09:50:15 pm: 1) not necessarily square units. it'll just be the unit of independant variable by the unit of dependant variable

in this case, time * population/time = population.
Title: Re: Properties of integrals
Post by: shinny on August 23, 2008, 10:47:00 pm: Not sure if this is a full mark solution, or even the best way to do it but it seems to do the job.
2.
$y^{n}=x \cdots1$

$y=x^{n}\cdots2$

$\Rightarrow y^{n}=(x^{n})^{n}\cdots3$

$\text{Equate 1 and 3} \Rightarrow (x^{n})^{n}=x$

$\Rightarrow x^{n^{2}}-x=0$

$\Rightarrow x(x^{n^{2}-1}-1)=0$

$\text{Therefore x=0 or } x^{n^{2}-1}-1=0$

$\Rightarrow x^{n^{2}-1}=1$

$\Rightarrow x=1$

$\text{Substitute into equation 1}$

$\text{Therefore y=1}$

$\text{Therefore for } n\in\mathbb{Z},\text{ both curves will always intersect at (1,1)}$

And why do i have a feeling this solution will be obsolete in comparison to someone elses. Looks a bit too ugly to me. Anyway back to before, still, I've been told MANY times that double differentiation to verify stationary points strictly isn't allowed since it isnt in the course and its unfair for spesh students to have any advantage? I think I'll just play it on the safe side...
Title: Re: Properties of integrals
Post by: /0 on August 23, 2008, 10:51:42 pm: Thanks mao and shinjitsuzx, I like that solution :)
Title: Re: Properties of integrals
Post by: nerd on August 25, 2008, 09:26:03 pm: Sorry to bring this up again, but I still don't understand the first 2 questiosn that DivideBy0 posted...can someone please explain them again?
Title: Re: Properties of integrals
Post by: shinny on August 25, 2008, 09:37:33 pm: I don't think those questions should be popping up in methods so you shouldn't need to worry. I really don't know how to explain them algebraically without using spesh knowledge, and its hard to provide a legit graphical solution given that you don't have an actual graph.