find the surface area generated when rotating y=x^2 from(1,1) to (2,4) about the x axis @hartnn

Question

hartnn · Answer

there's a formula for this right ?

anonymous · Answer

integral of 2pi (x or y) ds

hartnn · Answer

some thing like this ? 
$\Large \int  \limits_1^2x^2 \sqrt{1+4x^2}dx$
?

hartnn · Answer

oh, i missed 2pi ...and i didn't know that...

hartnn · Answer

how would you find ds ?

anonymous · Answer

ds = sqrt(1+ (dy/dx))...basically the sqrt you gave in your post

hartnn · Answer

$\Large 2\pi\int  \limits_1^2x^2 \sqrt{1+4x^2}dx$

anonymous · Answer

i was just a little confused about the points of integration but it's actually quite simple ( about x axis means use x coords and about y axis means use y coords right?)

hartnn · Answer

yes!

anonymous · Answer

should i take u=x or do this:
intgrl x^2 4(1/4+x^2) and then let u=1/4+x^2?

hartnn · Answer

thats not so easy to integrate....

hartnn · Answer

since i see the form x^2 +a^2 
i would have used the substitution of x = a tan t 
any better ideas any two of you have got ?

anonymous · Answer

i thought of letting u =1+4x^2 then du=8x =>8du=x but then the x^2 on the outside wouldn't be accounted for and i dont think theres such  a thing as du^2

hartnn · Answer

thats correct, it won't work

anonymous · Answer

i dont follow the tan substitution you have, could you explain it out to me?

hartnn · Answer

when we see the form x^2+a^2 in the integral, we plug in x = a tan t
so that x^2+a^2 becomes a^2 sec^2 t
just a trigonometric substitution to simplify things...

anonymous · Answer

ok, so we'll be integrating x^2(1sec^2t) ?

hartnn · Answer

but we need to put every x as a tan t 
and find dx too in terms of t and tan t
x= a tan t 
dx = .... ?

anonymous · Answer

a sec^2t

anonymous · Answer

a^2

hartnn · Answer

just ( a sec^2 t) dt 
don't forget the dt too
so now whats your integral in terms of 't' ?

anonymous · Answer

is it t=a tan^-1 x?

hartnn · Answer

....
so, your integral becomes 
$\large \int (a^2\tan^2 t)\sec t \: \:  a\sec^2 t dt$

got this ?

hartnn · Answer

this may not be the most efficient way to integrate this, but i can't think of other way...if any o2 of you can think, then you're welcome..

hartnn · Answer

here a = 1/2

anonymous · Answer

i'm still a little confused, i'll revise some more trig substitution and get back to you later

hartnn · Answer

ok, 
you can ask doubts related to this right now if you have any...
or you can get back later too...

anonymous · Answer

it's cool, i'll post again later if i still have questions

hartnn · Answer

good luck! :)