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5.d – Applications of Integrals

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

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5.d – Applications of Integrals

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  1. 5.d – Applications of Integrals

  2. 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?

  3. 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

  4. More Properties of the Definite Integrals Properties of Odd and Even: Suppose f is continuous on [– a, a].

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

  6. 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.

  7. 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.

  8. 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

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

  10. 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.

  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?

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