prove cross-section of elliptic paraboloid is a ellipse

chris_usyd

oh about the r drdθ..
that is deltA=r drdθ..

tiny-tim

ah, now i see

yes, that is a correct way of finding the volume …

you slice the volume into square vertical columns of height z = f(x,y) and base dxdy (and therefore volume x dxdy), and then sum all the individual volumes

so the volume is ∫∫ f(x,y) dxdy, = ∫∫ f(x,y)r drdθ

but it's not the only way … you can slice the volume other ways, which may be easier

in this case, since the previous parts of the question are all about the horizontal cross-sections,

i'm guessing that they intended you to use horizontal slices, find the area of the intersection of that circle-and-ellipse (the circle part is easy, the ellipse part fairly easy), and then integrate that area over z

(your solution looks ok, but i haven't checked right through it)

chris_usyd

thanks Tim, in part a) of this question, i found the semi-axes of this elliptic paraboloid are b*sqrt((h-z)/h) and a*sqrt((h-z)/h)
therefore the intersection of the ellipse at height z is gonna be [a*sqrt((h-z)/h)]*[b*sqrt((h-z)/h) ]*pi, isn't? which is just pi*(h-z)/h*a*b.
then how am i gonna use polar coordinates??? because a and b are constant numbers...

: )) tim its 11pm in sydney now, time for bed...
if u reply my post, i might not be able to read it tonight, but i will check tomorrow moring.. thanks for your help, have a good day!!

tiny-tim

you need to find the four values of θ where the circle and ellipse meet

over two sections, that's just the area of a sector of a circle, 1/2 r²(θ₁ - θ₂)

over the other two sections, it's the area of a sector of an ellipse, which is … ?

sleep tight!

TDR14

I'm confused about how you got the area?

PS I'm doing this same course and assignment too

tiny-tim

Hi TDR14! Welcome to PF!
There's various ways of doing it, but the simplest is to note that an ellipse is a squashed circle, by a factor b/a along the minor axis,

so the area is πa²*b/a, = πab

(technically, one proves this with the substitution y' = ay/b )

TDR14

ok, I know that.

But how do you put it in terms of x,y,z etc?

I got A=∏*(x/sqrt((h-z)/h))*(y/sqrt((h-z)/h))

Then I just follow chris_usyd's way of integrating and so on and so forth???

tiny-tim

i don't understand …

A does not depend on x or y, only on z, see chris's formula …

TDR14

ok I get it now, then how do you find volume by slicing?

tiny-tim

you find the area A as a function of z, then the volume is ∫ A dz