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5.d – Applications of Integrals. Indefinite Integrals and Area.

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Indefinite Integrals and Area

The definite integral is related to the area bound by the function f(x), the x-axis, 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 x-axis, 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].



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:


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



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


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?




A particle moves with a velocity v(t). What does

and represent?

t = a●

●t = b








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.



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?