Volume from Its Slices
How to find the volume of a figure from its slices: formula, 1 example, and its solution.
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.
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.
e0 = 1
-3 - 1 = -4
So e3 - 4 is the answer.