1.
The fundamental theorem of calculus
(In words, as much as possible)
David C, 2014
2.
The fundamental theorem of calculus says that the process of finding the area
under a curved graph line is mathematically equivalent to un-doing a
differentiation.
What does that mean?
I’ll try to show you in an informal, hand-waving sort of way.
3.
Suppose I use a straight line to approximate the slope between points x1 and x2.
12
12
xx
yy
m
y1
y2
x1 x2
4.
y1
y2
x1 x2
I could rearrange this as shown, for mysterious reasons that I’ll keep to myself.
1212 yyxxm
12
12
xx
yy
m
5.
Now any time you multiply two numbers together, like m and (x2-x1),
this is arithmetically the same as finding the area of a rectangle.
m
x1 x2
1212 yyxxm
6.
m
What could this possibly mean in real terms?
x1 x2
Now any time you multiply two numbers together, like m and (x2-x1),
this is arithmetically the same as finding the area of a rectangle.
1212 yyxxm
7.
m
Well this could be the area under a part of a curved graph line.
x1 x2
1212 yyxxm
But what could this graph possibly be?
8.
m
m(x).
x1 x2
1212 yyxxm
The height of the rectangle is m,
so it suggests that the vertical axis is m(x).
9.
m
Now remember that m(x) is a slope function,
so the area of the rectangle is kind of like
the area under the graph of a slope function,
especially if x2 and x1 are very close together.
m(x).
x1 x2
1212 yyxxm
10.
m
m(x).
Therefore the area of a very skinny rectangle
under the graph m(x) is the same as y2 - y1.
x1 x2
1212 yyxxm
11.
m
m(x).
And what is y2 - y1?
Well, think about it:
If m is the derivative of y,
then y is the anti-derivative of m.
x1 x2
1212 yyxxm
Therefore the area of a very skinny rectangle
under the graph m(x) is the same as y2 - y1.
12.
m
m(x).
x1 x2
So to get the area under part of a curvy line, you pretend that the area is a
rectangle and anti-differentiate the formula at the left and right edges of
the rectangle.
The problem, however, is that the rectangle has
to be very skinny if this approximation is to be
any good.
13.
m(x).
x1 x2
You can make it work, though, by putting a whole lot of rectangles
together, side-by-side.
The area of each rectangle is equal to
antiderivative( right edge ) - antiderivative( left edge )
14.
m(x).
x2
You can make it work, though, by putting a whole lot of rectangles
together, side-by-side.
The area of each rectangle is equal to
antiderivative( right edge ) - antiderivative( left edge )
When you add all the rectangles together, all those
antiderivatives cancel out except for the ones at the
far edges, left and right.
x1
15.
m(x).
x2
So the area under any graphline
is equal to the antiderivative of the left and right edges
of the region you’re looking at.
(This, in words, is the fundamental theorem.)
x1
16.
m(x).
9
An example of this theorem in action would be
finding the area under a line that curves according to, say, a cubic formula.
1
3
2xm
4
4
2
xy
Area between x=1 and x=9 is 4
4
24
4
2
19 3280
17.
m(x).
The traditional notation for this process is a little bit more ornate
but it means exactly the same thing.
9
1
3
2 dxxarea
9
1
4
4
2
x
...etc
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