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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 .
 

 
    1. 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:
 
 
    1. 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
 



M Ransom

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