1 5.d – Applications of Integrals. 2 Definite Integrals and Area The definite integral is related to the area bound by the function f(x), the x-axis,

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Presentation transcript:

1 5.d – Applications of Integrals

2 Definite 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.

3 Indefinite Integrals and Area Examples: 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 which cases do definite integrals yield actual area? Does definite integrals always yield actual area? Does the value of (d) represent the actual area bound by x 2 – 6x + 5, the x-axis, x = -1 and x = 7? Write two examples of definite integrals that will yield the actual area.

4 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

5 Examples 1. Use the properties of integrals to verify the inequality without evaluating the integrals.

Important: For the net change theorem to apply, the integrand must be a rate of change. 6 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 (x) from a to b. Must Be A Rate Of Change (1)

7 Examples 2. What does the integral below represent if v(t) is the velocity of a particle in m/s. 3. A honeybee population starts with 100 bees and increases at a rate of n(t). What does represent?

8 Examples 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?

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

10 Examples 6. 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 displacement during the given time interval. (c) The total distance traveled during the time interval.

11 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?