1 5.4 – Indefinite Integrals and The Net Change Theorem

2 Indefinite Integrals Use WolframAlpha to determine the following. Question: What does represent?

3 Indefinite Integrals In other words, F(x) is the _________________ of f (x).

4 Examples Evaluate each indefinite integral.

5 Definite Integrals where F(x) is the general antiderivative of f (x).

6 Examples

7 The Net Change Theorem The integral of a rate of change is the net change: Meaning: If F (x) represents a rate of change (m/sec), then (1) above represents the net change in F (m) from a to b. Must Be A Rate Of Change Important: For the net change theorem to apply, the function in the integral must be a rate of change. (1)

8 Examples 1.The current in a wire, I, is defined as the derivative of the charge, Q. That, is I(t) = Q(t). What does represent? 2. A honeybee population starts with 100 bees and increases at a rate of n(t). What does represent?

9 Examples 3. If f (x) is the slope of a trail at a distance of x miles from the start of the trail, what does represent? 4. If the units for x are feet and the units for a(x) are pounds per foot, what are the units for da/dx. What units does have?

10 Example A particle moves with a velocity v(t). What does and represent? |0|0 s(t)s(t) t = a ● ● t = b

11 Examples 1. The acceleration functions (in m/s 2 ) and the initial velocity are given for a particle moving along a line. Find (a) the velocity at time t and (b) the distance traveled during the given time interval.

12 Examples 2. Water flows from the bottom of a storage tank at a rate of r(t) = 200 – 4t liters per minute, where 0 ≤ t ≤ 50. Find the amount of water that flows from the tank in the first 10 minutes.