5.3 Definite Integralsand Antiderivatives

Greg Kelly, Hanford High School, Richland, Washington

Page 269 gives rules for working with integrals, the most important of which are:

2. 0a

af x dx If the upper and lower limits are equal,

then the integral is zero.

1. b a

a bf x dx f x dx Reversing the limits

changes the sign.

b b

a ak f x dx k f x dx 3. Constant multiples can be

moved outside.

1.

0a

af x dx If the upper and lower limits are equal,

then the integral is zero.2.

b a

a bf x dx f x dx Reversing the limits

changes the sign.

b b

a ak f x dx k f x dx 3. Constant multiples can be

moved outside.

b b b

a a af x g x dx f x dx g x dx 4.

Integrals can be added and subtracted.

b b b

a a af x g x dx f x dx g x dx 4.

Integrals can be added and subtracted.

5. b c c

a b af x dx f x dx f x dx

Intervals can be added(or subtracted.)

a b c

y f x

The average value of a function is the value that would give the same area if the function was a constant:

21

2y x

3 2

0

1

2A x dx

33

0

1

6x

27

6

9

2 4.5

4.5Average Value 1.5

3

Area 1Average Value

Width

b

af x dx

b a

1.5

The mean value theorem for definite integrals says that for a continuous function, at some point on the interval the actual value will equal the average value.

Mean Value Theorem (for definite integrals)

If f is continuous on then at some point c in , ,a b ,a b

1

b

af c f x dx

b a