Volumes -- parabolic cross sections --- problem with finding area of cross-section (1 Viewer)

[blackops23]

Hi guys, I'm having a problem with a question from Cambridge:

EX 6.3 Q7

Q. The base of a particular solid is x^2 + y^2 = 4. Find the volume of the solid if every cross-section perpendicular to the x-axis is a parabolic segment with axis of symmetry passing through the x-axis and height the length of the base.

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So how do I find the area of the parabolic cross-section? Obviously the width is delta(x) -- but I have no idea how to find the area. Other questions involved finding the area of right-angled isosceles triangles, equilateral triangles, squares, semi-circles etc....

But I've no idea what to do in the case of parabolic cross-sections..

Help would be greatly appreciated.

 

[kr73114]

Re: Volumes -- parabolic cross sections --- problem with finding area of cross-sectio

u could use trapezoidal rule because that is exact when finding area of parabola

 

[Hermes1]

Re: Volumes -- parabolic cross sections --- problem with finding area of cross-sectio

u could use trapezoidal rule because that is exact when finding area of parabola

isnt that simpsons which is exact?

 

[kr73114]

Re: Volumes -- parabolic cross sections --- problem with finding area of cross-sectio

isnt that simpsons which is exact?

yeh soz thas what i meant

 

[Hermes1]

Re: Volumes -- parabolic cross sections --- problem with finding area of cross-sectio

in exams they usually step u through finding the area of the parabolic cross-section. anyways ill give this question a go.

 

[blackops23]

Re: Volumes -- parabolic cross sections --- problem with finding area of cross-sectio

isnt that simpsons which is exact?

simpson's rule you say? so how would i apply it to this question? and how would i obtain the function values? i mean i don't even know what the function is lol (i.e. equation of parabola)...

 

[pwoh]

Re: Volumes -- parabolic cross sections --- problem with finding area of cross-sectio

^ area of parabola, then just do volume stuff as usual

You can use the Simpson's approximation for it as well, since Simpson's uses parabolas for approximation. But I don't remember what that was xD

 

[blackops23]

Re: Volumes -- parabolic cross sections --- problem with finding area of cross-sectio

^ area of parabola, then just do volume stuff as usual

You can use the Simpson's approximation for it as well, since Simpson's uses parabolas for approximation. But I don't remember what that was xD

Thanks mate, took me a while to understand what the hell the working out meant, but I got there in the end

So just to make sure -- that is the best/fastest/only way to find the equation of the parabola? And just out of curiosity, could someone please show me how to use simpson's rule for this particular question?

Thanks

 

[pwoh]

Re: Volumes -- parabolic cross sections --- problem with finding area of cross-sectio

Sorry it's quite messy xD

Fastest I could think of.
Generally I'd use general form (y = ax^2 + bx + c) but I skipped the bx term because it's clearly not shifted along x.

Looked up simpson's rule on wikipedia:

^ holy crap, magic

 

[Drongoski]

Re: Volumes -- parabolic cross sections --- problem with finding area of cross-sectio

Simpson's is based on fitting a quadratic (parabola) to 3 points. In this case the approximation coincides with the parabola (so not an approxn). That's why you get same answer with Simpson's in this case.

 

[pwoh]

Re: Volumes -- parabolic cross sections --- problem with finding area of cross-sectio

Simpson's is based on fitting a quadratic (parabola) to 3 points. In this case the approximation coincides with the parabola (so not an approxn). That's why you get same answer with Simpson's in this case.

Thanks Drongoski, the reason why I say 'magic' is because the solution is less the half the size of the first one xD

That said, I've never really had an intuitive understanding of Simpson's rule. You seem like someone who would know how to explain it - I mean, the algebra all works out but I can't seem to see how the '4' etc comes about geometrically.