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

11. Practice Time !!!
Find the total area between the curve y = 1 – x2
and the x-axis over the interval [0, 2].
A=
2
ò
1
2
0
1
1- x 2 dx = ò (1- x 2 ) dx + ò - (1- x 2 ) dx =
0
æ
x öù æ
x 3 öù
æ 1 ö éæ 8 ö æ 1 öù
ç x - ÷ú - ç x - ÷ú = ç1- - 0 ÷ - êç 2 - ÷ - ç1- ÷ú =
3 øû0 è
3 øû1 è 3 ø ëè 3 ø è 3 øû
è
3
1
2
2 æ 4ö 6
- ç- ÷ = = 2
3 è 3ø 3

12. The Mean Value Theorem for Integrals:
Over any interval, there exists an x value which creates a y value
that is the height of a rectangle which will equal the area under the
curve.
The Average Value:
The function value, f(c), found by the
Mean Value Theorem
1
Þ f ( c) =
b-a
b
ò f ( x)
a

13. Example
æ x 2 +1 ö
2 2æ
1ö
1
faverage =
ò ç x 2 ÷dx = 3 ò ç1+ x 2 ÷dx =
1 1è
ø
1 è
ø
22
2 2
2
2æ
1 öù
2 éæ 1 ö æ 1 öù
ç x - ÷ú = êç 2 - ÷ - ç - 2 ÷ú =
3è
x øû1 3 ëè 2 ø è 2 øû
2
2
2 éæ 1 ö æ 1 öù 2
êç1 ÷ - ç -1 ÷ú = × 3 = 2
3 ëè 2 ø è 2 øû 3

14. In analyzing the graph of F(x) we would look
at the derivative:
d
F ( x ) = ??? f (x)
dx
F ( x) =
x
ò (sint )dt = -cost ]a =-cos x - (-cosa) =
a
x
-cos x + cosa
d
d
F ( x) =
[-cos x + cosa] = sin x
dx
dx

15. The Fundamental Theorem of
Calculus, Part II
éx
ù
d
ê ò f (t)dt ú = f (x)
dx ë a
û

16. How about some
practice?
éx
ù
d
4
1. ê ò ( tan t ) dt ú =
dx ë 0
û
tan x
d éx 3 ù
2. ê ò t dt ú = x 3
dx ë 1
û
d é x sin t ù sin x
3. ê ò
dt ú =
dx ë 0 t
x
û
4

17. Integrals with Functions as
Limits of Integration
ég( x)
ù
d
ê ò f (t)dt ú = f ( g ( x )) × g' ( x )
dx ë a
û

18. Let’s Practice !!!
x2
d
dx
d
dx
ò
2
sin x
ò
1
1
1
× 2x = 1 × 2x = 2
dt = 2 3
3
t
x )
x6
x5
(
1- t ) dt = 1- sin 2 x × cos x = cos2 x × cos x = cos3 x
(
2
(
)