1. Properties of Definite Integrals

2. 1.
Reversing the limits changes the sign. 2.
If the upper and lower limits are equal, then the integral is zero. 3.
Constant multiples can be moved outside.

3. Reversing the limits changes the sign.1.
If the upper and lower limits are equal, then the integral is zero. 2.
Constant multiples can be moved outside. 3. 4.
Integrals can be added and subtracted.

4. 4.
Integrals can be added and subtracted. 5.
Intervals can be added
(or subtracted.)

5. IF f(x) is EVEN and the bounds are opposites.6.
IF f(x) is ODD and the
bounds are opposites. 7.
Recall: f(x) is EVEN if f(-x) = f(x)
f(x) is ODD if f(-x) = -f(x)

6. Average Value

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

9. Application problem: the speed of sound
At different altitudes in the Earth’s atmosphere, sound travels at different speeds. The speed of sound s(x), in meters per second, can be modeled by:
s(x) =
Where x is the altitude in km. What is the average speed of sound over the interval [0, 80]?