Introduction Introduction
What is an Antiderivative? (Indefinite Integral)
An antiderivative (aka an indefinite integral) is the opposite of a derivative.
The antiderivative of the function
f
is
F
, such that
F'(x) = f(x)
.
For example, the antiderivative of
f(x) = 3x^2
is
F(x) = x^3
.
(x^3 + 1)
,
(x^3 - \pi)
, and all other forms of
x^3
summed with a constant are also antiderivatives of
f(x) = 3x^2
.
The functions below share a derivative, as they have the same slope at each point:
Thus, we append antiderivatives with
+C
, which represents the integration constant.
The true antiderivative of
f(x) = 3x^2
is
F(x) = x^3 + C
.
What is an Integral? (Definite Integral)
An integral is the area under a curve. We find integrals using antiderivatives. (In other words, we find definite integrals using indefinite integrals.)
f(x) = x^2
The highlighted area represents the area under
f
on
(-1.5, 1.5)
:
To find the area under
f
, we find its antiderivative and evaluate it at the bounds of the integral.
In other words,
The area under f
on (-1.5, 1.5)=F(1.5) - F(-1.5)
.
\begin{align*}
F(x) &= \frac{1}{3}x^3 + C \\\\
F(1.5) - F(-1.5) &= [\frac{1}{3}(1.5)^3 + C] - [\frac{1}{3}(-1.5)^3 + C] \\\\
&= 1.125 + 1.125 \\\\
&= 2.25
\end{align*}
The area under
f
on
(-1.5, 1.5)
is
2.25 \text{ units}^2
.
Note that when evaluating integrals, the integration constant
+C
always cancels out.
Accumulation of Change
The area under a curve of a derivative is equal to the change in the original function.
The area under
f'
on
(a, b) = f(b) - f(a) =
the change in
f
from
a
to
b
.
This is a graph of
f'
. The red is the area on
(a,b)
.
Since the area below the x-axis is larger than the area above the x-axis,
f
decreased from
a
to
b
.