The Fundamental Theorem of Calculus The Fundamental Theorem of Calculus

Introduction

The Fundamental Theorem of Calculus establishes that differentiation and integration are inverse operations.

It has two parts.

Part 1: The Derivative of an Integral

Part 1 of the FTC states that the derivative of the integral of a function is the original function itself.

In mathematical terms (

a
is a constant):
\frac{d}{dx} \left[\int_{a}^{x} f(t) \, dt \right] = f(x)
The integral is the accumulated area under
f(t)
from
a
to
x
.

The derivative of this accumulated area with respect to
x
is the rate at which the accumulated area changes at
x
.

f(x)
is also the rate of change of the accumulated area at
x
.

Therefore, the derivative of the integral of
f(x)
is
f(x)
.

Part 2: The Integral of a Derivative

Part 2 of the FTC states that the integral, from

a
to
b
, of the derivative of a function is equal to the change in the original function.

In mathematical terms:
\int_{a}^{b} f'(x) \, dx = f(b) - f(a)

The derivative of
f(x)
is the rate of change of
f(x)
.

The integral of this rate of change from
a
to
b
is the total change in
f(x)
from
a
to
b
.

Therefore, the integral of the derivative of
f(x)
is the change in
f(x)
from
a
to
b
.