Finding Volume from Its Slices
How to find the volume from its slices: formula, examples, and their solutions.
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.
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)
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
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.
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.