# Volume from Its Slices

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

## 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):

∫_{a}^{b} S(x) dx.

## Example

Roughly draw the figure

with the h-axis.

At h = h,

the cross sectional area is

S(h) = e^{h} - 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

∫_{0}^{3} (e^{h} - 1) dh.

Solve the integral.

Definite Integral: How to Solve

The integral of e^{h} is itself: e^{h}.

Integral of e^{x}

The integral of -1 is -h.

Integral of a Polynomial

Put 3 and 0

into e^{h} - h.

e^{0} = 1

Zero Exponent

-3 - 1 = -4

So

e^{3} - 4

is the answer.