5.d – Applications of Integrals. Indefinite Integrals and Area.
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5.d – Applications of Integrals
Indefinite Integrals and Area
The definite integral is related to the area bound by the function f(x), the xaxis, and the lines x = a and x = b. definite integrals do not always yield area since we know that definite integrals can give negative values.
Examples a  c: Compute the definite integrals using your graphing calculators. Then compute the area bound by the graphs of the integrands, the xaxis, and x = a and x = b. In what cases do definite integrals yield actual area?
More Properties of the Definite Integrals
2. If f (x) ≥ 0 for a ≤ x≤ b, then
3. If f (x) ≥ g (x) for a ≤ x≤ b, then
4. If m≤ f (x) ≤ M for a ≤ x≤ b, then
More Properties of the Definite Integrals
Properties of Odd and Even: Suppose f is continuous on [– a, a].
Examples
1. Use the properties of integrals to verify the inequality without evaluating the integrals.
The Net Change Theorem
The integral of a rate of change is the net change:
(1)
Must Be A Rate Of Change
Important: For the net change theorem to apply, the integrand must be a rate of change.
Meaning: If f (x) represents a rate of change (m/sec), then (1) above represents the net change in f (x) from a to b.
3. A honeybee population starts with 100 bees and increases at a rate of n(t). What does
represent?
Examples
2. What does the integral below represent if v(t) is the velocity of a particle in m/s.
4. If f (x) is the slope of a trail at a distance of x miles from the start of the trail, what does
represent?
5. 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?
Examples
Example
A particle moves with a velocity v(t). What does
and represent?
t = a●
●t = b
s(t)

0
________
______________
Examples
6. The acceleration functions (in m/s2) and the initial velocity are given for a particle moving along a line. Find (a) the velocity at time t and (b) the displacement during the given time interval. (c) The total distance traveled during the time interval.
Examples
7. Water flows from the bottom of a storage tank at a rate of r(t) = 200 – 4t liters per minute, where 0 ≤ t ≤ 50. (a) Find the amount of water that flows from the tank in the first 10 minutes. (b) How many liters of water were in the tank?