# 5.3 Definite Integrals and Antiderivatives

```5.3 Definite Integrals
and Antiderivatives
Greg Kelly, Hanford High School, Richland, Washington
Page 269 gives rules for working with integrals, the most
important of which are:
1.
2.
3.

b
a

a

b
a
a
f  x  dx   f  x  dx
a
b
Reversing the limits
changes the sign.
f  x  dx  0 If the upper and lower limits are equal,
then the integral is zero.
k  f  x  dx  k  f  x  dx Constant multiples can be
b
a
moved outside.


1.
2.
a

a

b
a
3.
4.
b
a
f  x  dx   f  x  dx
a
Reversing the limits
changes the sign.
b
f  x  dx  0 If the upper and lower limits are equal,
then the integral is zero.
k  f  x  dx  k  f  x  dx Constant multiples can be
b
a

b
a
moved outside.
 f  x   g  x  dx  f  x  dx   g  x  dx
a
a
b
b
subtracted.


4.
b
a
 f  x   g  x  dx  f  x  dx   g  x  dx
a
a
b
b
subtracted.
5.
 f  x  dx  f  x  dx   f  x  dx
b
c
c
a
b
a
y  f  x
a
(or subtracted.)
b
c

The average value of a function is the value that would
give the same area if the function was a constant:
A
5
3
0
1 2
x dx
2
4
3
27
1 3

 x
6
6 0
3
9

 4.5
2
2
1.5
4.5
Average Value 
 1.5
3
1
0
1
y
2
1 2
x
2
3
Area
1 b
Average Value 

f  x  dx

a
Width b  a

The mean value theorem for definite integrals says that
for a continuous function, at some point on the interval the
actual value will equal to the average value.
Mean Value Theorem (for definite integrals)
If f is continuous on  a, b then at some point c in  a, b ,
1 b
f c 
f  x  dx

a
ba
p
```