How would i find what a and b are?

Question

Set up an integral that computes the volume of a cone of height 12 whose base is an ellipse with semi-major axis 6 and semi-minor axis 4. The area formula for an ellipse is pi(a)(b), where a is the length of the semi-major axis and b is the length of the semi-minor axis.

Answer 1

Think of the cone as a "stack" of ellipses of height 12. Use x to represent the distance from a given slice to the vertex of the cone. So the area of each slice will be pi (6*x/12) (4*x/12) =pi x^2 / 6

So volume of cone will be the integral from 0 to 12 of pi x^2 / 6 dx

Answer 2

Now if the problem was just a basic Geometry's problem the calculation of the volume of the cone would be

is easy to be found since V = 1/3 (area of base) ( height) = (1/3) (π⋅6⋅4)⋅12 = 96 π cubic units

But we need to use Calculus to reach the same result.

.

We introduce a rectangular coordinate system in which the x-axis passes through the apex of the cone.

is perpendicular to the base , and passes through the center of the ellipse of the elliptical base.

Also the apex of the cone is at the origin

At any x in the interval [0,12] on the x-axis , the cross section perpendicular to the x-axis is an ellipse similar to the ellipse of the base of the cone .

If we let A to denote the Area of the ellipse and A_x the area of the perpendicular slice of the cone at x

on the x-axis then using similarity

A / A_x = ( 12 / x )² (See footnote)

Then A_x= ( x/ 12 )² A

⋅A_x= ( x/ 12 )²⋅π⋅6⋅4

⋅A_x= ( x/ 12 )²⋅24π

A_x= π⋅x²/6

and the volume V of the cone is

V= ∫¹²₀ A_x dx = ∫¹²₀ π⋅x²/6 dx =(π/6) ∫¹²₀ x² dx = 96 π

Footnote: When to geometric figures are similar the ratio of their areas is equal to the square of their

rate of similarity.

Answer 3

Diana, write me a message so I can send you a picture of the proper setup for the cone, which you would use to set up a Reimann sum, and then get an integral you can use to calculate the volume.

You set up your bottom level as a 'zero', then take a little slice of the cone. The volume of the slice would be ∆h*π*α*β where delta-h is the height of this cylinder and alpha and betta are the corresponding axes for that "cut."

Let hi be the height for that cylinder from the bottom (which means the coordinate of the top is 12).

You express α and β through "proportions of a triangle":

α / a = (12 - hi) / 12 and β / b = (12 - hi) / 12

That little volume then is :

∆h * π * a * [ (12 - hi) / 12 ] * b [ * (12 - hi) / 12 ] =

1/144 * ∆h * π * a * b * (12 - hi)^2 =

1/6 * ∆h * π * a * b * (12 - hi)^2

Here you set up a Reimann sum, take your delta-h to zero and get the integral. It is very overwhelming to type it, so I can just send you a pic.