Calculus Calculating Definite Integrals
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In the Area
Calculating Definite Integrals

 

 How do we calculate the area between a curve and the X axis

 from one X value to another?
 
 Say we want the area between the line F(X) = X 2 and the X 
 axis from X = 0 to X = 1.
 We set up an integral notation and put the smaller number at the bottom 
 and the bigger number at the top.

 

 Find the integral of X 2dX, 
 and place it in square brackets.

 

 The first step is to substitute the top number in for X
 

 

 
 The last step is to substitute the lower number into the term
 and subtract that from what we already have.
 

 

 
 And we're back to the same value, 1/3.
 
 OK, now lets find the area between the F(X) = X 2 curve and the X axis 
 from X = 5 to X = 20 ...
 
 
 
 
 
 
 
 
 
 One nice thing about F(X) = X 2, is that it never crosses the X axis.
 

 

 
 What happens if we have an equation that does cross the X axis?
 
 What if we have an equation like:

 F(X) = X3

 
 And we want to find the area between the curve
 and the X axis, from X = -2 to X = 2?

 

 
 If we just plug away, we get:
 
 
 
 
 
 
 
 
 
 OOPS!
 I can see we have some area here,
 how come we got zero for an answer?
 
 When left on their own, 
 the equations will see any area below the axis as NEGATIVE area, and subtract it.
 
 If there is more area below the axis, 
 the whole answer will be negative.
 
 OK, How come the area below the X axis is negative area?
 
 Let's talk about how the equations "build" the area.
 
 Say we have the equation:

 F(X) = X2

 
 The graph looks like this:

 

 
 The area under the graph from X = 0 to X = 3
 looks like this:

 

 The graph of the equation

 F(X) = X2 + 5

 
 Looks like this:

 

 
 The thing is, the area between the curve and the X axis is made up of two pieces.
 One is the same odd shape area we had with F(X) = X 2.
 The other is a rectangle 5 units high and 3 units wide.

 

 
 When we find the integral,
 we even wind up with these two area pieces as separate terms.
 

 

 
 The graph of an equation like F(X) = X 2 - 5, looks like this:
 

 

 
 The area between this equation line and the X axis, 
 from X = 0 to X = 3 is

 

 The area is:

 

 The way the equation builds this is:

 

 
 So how do we get all of the area parts to give us positive numbers?,
 
 OK, say we have this equation ...

 F(X) = X2 - 1

 
 and we want to find the total of all the areas between this curve 
 and the X axis from X = -2 to X = +2

 

 
 Say we want the total area, including all of these three pieces.
 That is, we want to count them all as positive areas.
 
 OK, so we know that if we just run the numbers, 
 the middle piece will come out as negative.
 To do this right, we need to find out the X values 
 where that middle piece starts and ends.
 Those are the places it crosses the X axis.
 AND AT THOSE PLACES THE FUNCTION EQUALS ZERO!
 
 So set the function equal to zero and solve for X ...
 
F(X) =   X2 - 1
0 =   X2 - 1
X2 =   1
X =   1

 So we have three sections, X = -2 to X = -1, X = -1 to X = 1, And X = 1 to X = 2.

 Two of these are above the X axis and one is below it.

 To make the area of the one below the X axis work out positive,

 we need to subtract it.
 That makes the whole thing look like this:

 

 

 
 There are a whole lot of places where you could make a mistake
 on something like this,
 SO BE CAREFUL!
 Losing track of a sign is a VERY COMMON MISTAKE
 
 The fact that all of the numbers in that last step were either plus or minus 2/3 ,
 probably has some cosmic significance.
 But we really aren't worried about that right now.
 

   copyright 2005 Bruce Kirkpatrick

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