Definite Integral Across Addition, Subtraction

Date: 8/10/96 at 15:8:37
From: Anonymous
Subject: Definite Integral Across Addition, Subtraction

What is the integral of 3(x^2)-5x+9 from 0 to 7?

Date: 8/11/96 at 18:32:14
From: Doctor Paul
Subject: Re: Definite Integral Across Addition, Subtraction

We first need to know a property of integrals:

 /                         /           /
 | [ a(x) + b(x) ] dx =    | a(x) dx + | b(x) dx
 /                         /           /

So we can rewrite your integral from:

 /7
 |  3x^2 - 5x + 9 dx
 /0

to:

 /             /        /
 | 3x^2 dx +  | 5x dx + | 9 dx
 /            /         /
 
                   /                    /
now remember that | a * b(x) dx   =   a | b(x) dx
                  /                     /


so let's pull the constants out in front:

  /            /          /
3 | x^2 dx + 5 | x dx + 9 | dx
  /            /          /

                          ^^^^^
                          note that we really have x^0 here..

Now you need to know how to integrate:

If you integrate x^n dx then you get [ (1/(n+1)) * x^(n+1) ]

Let's perform the integration:

3*[ (1/3) * x^3 ] + 5*[ (1/2) * x^2) ] + 9*[ (1/1) * x^1) ]

Simplify it out..

x^3 + (5/2)x^2 + 9x

since we're looking for a definite integral we don't need to add a 
constant of integration (the '+ C' at the end)

let's apply the limits:

                   |7
x^3 + (5/2)x^2 + 9x|
                   |0

Now we use the Fundamental Theorem of Calculus to solve this..

/b
|  f(x) dx  =  F(b) - F(a)  where F(x) is *any* antiderivitive of f(x)
/a
 
The word *any* in the definition above allows us to choose our 
constant of integration to be zero in these definite integral 
problems.

Let's solve it. We already know F(x).

F(x) = x^3 + (5/2)x^2 + 9x

F(7) - F(0)  =  283.5 - 0  =  283.5

I hope this helps clear things up..

Regards,

-Doctor Paul,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/