1. The Definite Integral
and The Fundamental
Theorem of Calculus

2. This method used the sum of
the area of intervals under a
curve- called Reimann Sums

3. The limit of the sums of intervals is the same as
a definite integral over the same interval.

6. • A’ (x) = f (x)
• A (a) = 0
and F (x) = A (x) + C
• A (b) = A
A (x)
b
F ( b ) - F ( a ) = é A ( b) + C ù - é A ( a ) + C ù =
ë
û ë
û
A ( b) - A ( a ) =
A-0 =
A

7. The Fundamental Theorem
of Calculus, Part I
b
ò
a
f ( x ) dx = F(x)] a
b

8. How about some
practice?
ù
3
3
3
3
2x ú
2 æ 2 2 ö 2 æ 9 2 -12 ö = 2 27 -1 = 52
x dx = ò x dx =
÷
(
)
= ç 9 -1 ÷ = ç
ú
3è
3
ø 3
3è
3
1
ø
ú1
û
9
1. ò
9
1
p
3 9
2
1
2
p
ù
cos x ù 2
sin x
1é æp ö
= - êcos ç ÷ - cos0ú = - 1 [ 0 -1] = 1
2. ò
dx = ú
5 û0
5
5ë è 2 ø
û
5
5
0
2
ù = 5 ( eln3 - e0 ) = 5 (3-1) =10
3. ò 5e dx = 5e û0
ln3
0
x
x ln3

9. More Examples !!!
6
Evaluate
ò f (x)dx If
0
2
ò
0
ì x 2,
ï
f (x) = í
ï3x - 2,
î
x<2
x³2
ù
x ù 3x 2
2
x dx + ò (3x - 2) dx =
- 2x ú =
ú +
3 û0 2
û2
2
6
æ8 ö
128
ç - 0 ÷ + ( 42 - 2) =
è3 ø
3
3 2
6

10. TOTAL AREA
A1
a
A3
A5
A2
Total area =
A4
b
ò
a
f (x) dx
b