Finding Volume from Its Slices

Finding Volume from Its Slices

How to find the volume from its slices: formula, examples, and their solutions.

Formula

V = [inteval, from a to b, of S(x) dx], S(x): Area of its slice

The volume of a 3D figure is
the integral of its cross-sectional area: S(x).
(cross-sectional area = sliced area)

So V = ∫ab S(x) dx.

Example 1

Water is filled into a tank below. If the level of water is h, then the surface area of water is e^h - 1. Find the volume of water in the tank when the level of water is 3.

When the height is h,
S(h) = eh - 1.

And h changes in [0, 3].

So V = ∫03 (eh - 1) dh.

Solve the integral.

Indefinite integration of ex

Example 2: Proof of the Formula (Volume of a Cone)

Show that the volume of the given cone is 1/3*pi*r^2*h.

To find the cross-sectional area S(y),
find the radius of the cross-sectional circle (brown).

If the height is h,
then the radius of the cross-sectional circle is r.

If the height is (h - y),
then the radius of the cross-sectional circle is (brown).

Then the big right triangle and the small right triangle
are similar,
because they show the AA similarity:
(Angle) Both triangles have a right angle.
(Angle) Both triangles have the same top angle.

So set a proportion
between the sides of these two right triangles:
(h - y)/(brown) = h/r.

Similarity of sides in triangles

Then (brown) = r(1 - y/h).

S(y) = π⋅(brown)2

Area of a circle

And (brown) = r(1 - y/h).

So S(y) = π[r(1 - y/h)]2.

Square of a sum

S(y) = πr2(1 - [2/h]⋅y + [1/h2]⋅y2).

And the height changes in [0, h].

So V = ∫0h πr2(1 - [2/h]⋅y + [1/h2]⋅y2) dy.

Solve the integral (with respect to y).

Definite integration of polynomials

Cancel the dark gray terms.

Then V = (1/3)πr2h.

So this is the proof of the volume of a cone formula.