# Fundamental Theorems of Calculus

## Overview

In simple terms these are the fundamental theorems of calculus:

1

Derivatives and Integrals are the inverse (opposite) of each other.

2

When we know the indefinite integral:

F =
f(x) dx

We can then calculate a definite integral between a and b by the difference between the values of the indefinite integrals at b and a:

b
a
f(x) dx = F(b) − F(a)

Let's explore the details:

## First Fundamental Theorem of Calculus

For a continuous function f(x) on an interval [a, b] with the integral:

F(x) =
x
a
f(t) dt

then the derivative of the integral F(x) gets us the original function f(x) back again:

F'(x) = f(x)

This means that the derivative of the integral of f with respect to its upper limit is the function f itself.

### Example: f(x) = 2x

The integral of 2x is x2, and using the second theorem (below):

F(x) =
x
a
2t dt = x2 − a2

Taking the derivative:

F'(x) = ddx(x2 − a2) = 2x − 0 = 2x

So the derivative of the integral of 2x got us 2x back again.

### Example: Constant speed

A car travels at a constant speed of 50 km per hour for exactly one hour:

Speed:
50 km per hour
Integral of 50 km per hour for one hour:
50 km
Derivative of 50 km over one hour:
50 km per hour

## Second Fundamental Theorem of Calculus

When we have a continuous function f(x) on an interval [a, b], and its indefinite integral is F(x), then:

b
a
f(x) dx = F(b) − F(a)

In other words the definite integral of f(x) from a to b equals the difference in the values of F(x) at b and a

This makes calculating a definite integral easy if we can find its indefinite integral.

### Example

Imagine we're filling a tank with water, and the rate at which water flows into the tank is given by f(t), where t is time in minutes.

If F(t) measures the total volume of water in the tank at any time t, then the amount of water added to the tank between times a and b is F(b) - F(a).

f(t):
t2
F(t):
t3/3
So the amount of water added to the tank between 3 and 6 minutes is
6
3
f(t) dt:
F(6) − F(3)
= 63/3 − 33/3
= 63