Volume from Its Slices

How to find the volume of a figure from its slices: formula, 1 example, and its solution.

Formula

Formula

To find the volume of a 3D figure,

add up the cross sectional slices.

If the cross sectional area is S(x),
then the volume of each slice is
S(x)⋅dx = S(x) dx.

So the volume of the figure is
the integral of the cross sectional area S(x):
ab S(x) dx.

Example

Example

Solution

Roughly draw the figure
with the h-axis.

At h = h,
the cross sectional area is
S(h) = eh - 1.

It says to find the volume when h = 3.
So the height of the figure is 3.

So the volume of a figure is
03 (eh - 1) dh.

Solve the integral.

Definite Integral: How to Solve

The integral of eh is itself: eh.

Integral of ex

The integral of -1 is -h.

Integral of a Polynomial

Put 3 and 0
into eh - h.

-3 - 1 = -4

So e3 - 4 is the answer.