Fundamental theorem of calculus
Given any antiderivative F the area
under the curve between a and b
is given by the
definite integral
( )
( )
( )
b
a
f x dx
F b
F a
=
−
∫
Why is this true?
Area(a,b+h) - Area(a,b)
f(b) h so
Area(a,b+h) - Area(a,b)
( )
h
f b
≈
→
Problem
3
0
Find
(8
2 )
x dx
+
∫
Area under f(x) = 8 + 2x
from a=0 to b=3
3
0
2
(8
2 )
(3)
(0)
( )
8
(3)
24
9
33
(0)
0
Area =
(3)
(0)
33
x
dx
F
F
F x
x
x
F
F
F
F
+
=
−
=
+
=
+
=
=
−
=
∫
Area under f(x) = 8 + 2x
from a=0 to b=3
1
2
1
2
Trapezoid with height
3,
8,
14
8
14
Area
3
33
2
2
h
b
b
b
b
h
=
=
=
+
+
=
=
=
8
14
0
3
Porsche 928, p. 415
The velocity at t
seconds is
10t – t
2
/4
for t
≤
20 seconds
How far does the car
move during this
time?
1 mile per hour
≈
½ yard per second
0
20
40
60
80
100
0
5
10
15
20

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