# Fundamental Theorems of Calculus

## Overview

In **simple terms** these are the fundamental theorems of calculus:

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

When we know the indefinite integral:

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:

Let's explore the details:

## First Fundamental Theorem of Calculus

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

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

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 x^{2}, and using the second theorem (below):

^{2}− a

^{2}

Taking the derivative:

^{2}− a

^{2}) = 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:

**50 km per hour**for one hour:

**50 km over one hour**:

Note: "C" is however far the car had traveled already.

## 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:

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)*.

^{2}

^{3}/3

^{3}/3 − 3

^{3}/3