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.
a a
b
a
f x dx f x dx
a b
Reversing the limits changes the sign.
2.
f x dx 0
If the upper and lower limits are equal, then the integral is zero.
b
3.
b
a
k f x dx k f x dx
a
Constant multiples can be moved outside.
1. 2.
a a
b
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.
b
b
3.
a
k f x dx k f x dx
a
Constant multiples can be moved outside.
b b
4.
b
a
f x g x dx f x dx g x dx a a
Integrals can be added and subtracted.
4.
b
a
f x g x dx f x dx g x dx a a
b b
Integrals can be added and subtracted.
5.
f x dx f x dx f x dx
b c c a b a
y f x
Intervals can be added (or subtracted.)
a
b
c
The average value of a function is the value that would give the same area if the function was a constant:
5 4 3 2
A
3
3
0
1 2 x dx 2 9 4.5 2
27 1 3 x 6 6 0
1.5
1
4.5 Average Value 1.5 3
1 2 3
0
1 2 y x 2
Area 1 b Average Value a f x dx 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 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 a f x dx ba
p