Suppose I have a solid like y=1-x^2 and y=0 rotated about x....

frank789

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hi all

suppose I have a solid like y=1-x^2 and y=0 rotated about x.

if i cut disks parallel to x, each disk has a radius of its x co-ordinate and a thickness of del y

summings those disks across the y in which they exist should have the integral

V = int{-1,1}pi(x)^2 dy

or

V = int{-1,1}pi((1-y)^1/2)^2 dy

if i cut disks parallel to y axis the have radius of the y co-ordinate and thickness of del x

v=int{-1,1}pi(y)^2 dx

or

v=int{-1,1}pi(1-x^2)^2 dx

why am I getting two different volumes? conceptually i dont understand...am i not still summing individual disks across a region?
thanks!
 
hi all

suppose I have a solid like y=1-x^2 and y=0 rotated about x.

if i cut disks parallel to x, each disk has a radius of its x co-ordinate and a thickness of del y

summings those disks across the y in which they exist should have the integral

V = int{-1,1}pi(x)^2 dy

or

V = int{-1,1}pi((1-y)^1/2)^2 dy

if i cut disks parallel to y axis the have radius of the y co-ordinate and thickness of del x

v=int{-1,1}pi(y)^2 dx

or

v=int{-1,1}pi(1-x^2)^2 dx

why am I getting two different volumes? conceptually i dont understand...am i not still summing individual disks across a region?
thanks!

If you cut parallel to the x-axis, you don't get discs!

Your first method finds the volume when the region is rotated about the y-axis, which is an entirely different problem. (Your limits of integration are also wrong.)

There is a way to integrate with respect to y for this volume (cylindrical shells), but I wouldn't want to use it.
 
addition to dr peterson

hi all

suppose I have a solid like y=1-x^2 and y=0 rotated about x.

if i cut disks parallel to x, each disk has a radius of its x co-ordinate and a thickness of del y

summings those disks across the y in which they exist should have the integral

V = int{-1,1}pi(x)^2 dy

or

V = int{-1,1}pi((1-y)^1/2)^2 dy

if i cut disks parallel to y axis the have radius of the y co-ordinate and thickness of del x



v=int{-1,1}pi(y)^2 dx

or

v=int{-1,1}pi(1-x^2)^2 dx

why am I getting two different volumes? conceptually i dont understand...am i not still summing individual disks across a region?
thanks!

dr peterson said.
If you cut parallel to the x-axis, you don't get discs!

my addition
you don't get circles parallel to the x axis, you get a 'parabolic shaped' ellipse. [Imagine the parabola rotated towards and away from you 90 degrees, then imagine the shape the apex makes about y axis.] you can get 'slices', but they are not round.

if the shape were a semicircle rotated about the x, you could sum 'discs' parallel to x or y and get a sphere.
 
dr peterson said.
If you cut parallel to the x-axis, you don't get discs!

my addition
you don't get circles parallel to the x axis, you get a 'parabolic shaped' ellipse. [Imagine the parabola rotated towards and away from you 90 degrees, then imagine the shape the apex makes about y axis.] you can get 'slices', but they are not round.

if the shape were a semicircle rotated about the x, you could sum 'discs' parallel to x or y and get a sphere.

I'm not entirely sure what "'parabolic shaped' ellipse" means; but you are right that it would take considerably more work to decide exactly what shape each slice would have, and then its area.

Incidentally, I would prefer to talk about slicing perpendicular to the x-axis, rather than parallel to the y-axis, since this is ultimately a three-dimensional depiction. This makes it clearer to me.
 
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