volume questions (cross section thing) (1 Viewer)

[sasquatch]

The base of a particular solid is x² + y² = 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 its base.

This question was so hard..took me so long and i got 2pi. Apparently the answer is 256/9... i must have done something seriously wrong..hehe.

Please lend a hand. THanks.

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I have another question, but im sure(ish) that the book is wrong..well i dunno i got double the answer.

The base of a particular solid is the circle x² + y² = 4. Find the volume of the solid if every cross section perpendicular to the x-axis is a semi-ellipse with minor axis in the base of the solid and semi-major axis equal to its minor axis.

so minor axis = 2b = 2y, b = y.
semi-major axis = a = 2b = 2y

/\V ~= pi(ab)/\x
= 2pi y²/\x
= 2pi(4-x²)/\x

V = lim(x->0) sum(-2,2) 2pi(4-x²)/\x
= 2pi int(-2,2)(4-x²)dx
= 4pi int(0,2)(4-x²)dx
= 4pi [4x - (1/3)x³]²₀
= 64pi/3 cubic units

So what could i have done wrong

 

[bboyelement]

for the first one use the simpson's rule because its accurate for parabolas ... use 3 function values with the first is 0 and last is 0 and the mid is 2y

the second one you should find the are of an normal ellipse ... i remember this from the arnolds book ... A = pi*a*b ...you can find this by finding 2y of an ellipse with equation x^2/a^2 + y^2/b^2 = 1 then Area just integrate 2y with the limit of a to -a

once you got the area of an ellipse ... divide that by 2 hence A=(pi*a*b)/2 then volume should be easy to find

 

[sasquatch]

for the second one thats what i did... but i got 2piy² for the area. oh yeah shit.. i didnt divide by two :| damnit.... thanks!

 

[Sober]

I just solved the first one for 256/9, let me know if you still need help with it.

 

[STx]

I just solved the first one for 256/9, let me know if you still need help with it.

can you please post your solution?

 

[Sober]

It might have made more sense if I hadn't used the redundant variable h and had used z instead of reusing y but it should make a bit of sense.

 

[sasquatch]

hey thanks sober.. i like that method better than using the simpsons rule. thanks.

 

[STx]

Click to enlarge (12kB)

It might have made more sense if I hadn't used the redundant variable h and had used z instead of reusing y but it should make a bit of sense.

I get all of it except how you got the equation of the parabolic slice, whats the method for that?

 

[Sober]

It has the two roots h/2 and -h/2 so the porabola must be in the form of:

y = k(x-h/2)(x+h/2)

But the coordinate (0,h) must satisfy it so solve for k.