Introductory Calculus: Anti-derivatives, Integrals, Area Under a Curve

Given a function , an anti-derivative is another function which has as its derivative.

For example,

 

is an anti-derivative of .

 

In fact, the 7 is not unique. Any function of the general form

 

is an anti-derivative of .

 

In this lesson we will find anti-derivatives and in some cases find the exact value of . We will also use anti-derivatives to calculate the areas of regions bounded by the graphs of given functions.

 

For x to any power we have the following rule:

Anti-derivatives

if and , then has an anti-derivative with the general form           if , then has an anti-derivative with the general form

 

Let's look at some examples:

 

1. Find an anti-derivative of .

2. Find the area below the graph of , bounded below by the x–axis, on the right by , and on the left by .

 

We start by finding an anti-derivative, then substitute 3 and -1 for , evaluate, and subtract their values.

 

We know that our general anti-derivative is of the form

 

 

Substituting in 3 and -1 for x and subtracting gives us [notice that the "C's" cancel]

 

 

The area under the curve is 40/3 or 13.3 as shown on the diagram below.

 

 

Mathematically we write:

 

 

3. Find the anti-derivative of if we know that F(x) = 5 when .

 

We know that our general anti-derivative is of the form

 

 

To find our specific value for C, we set x = 1 and F(x) equal to 5 and solve for C:

 

 

Therefore the anti-derivative we want is