Definite Integral as Accumulated Change for AP Calculus

Practice problems for these concepts can be found at: Applications of Definite Integrals Practice Problems for AP Calculus

Business Problems

Profit = Revenue – Cost P (x) = R(x ) – C(x )
Revenue = (price)(items sold) R(x) = P (x )
Marginal Profit P'(x)
Marginal Revenue R'(x)
Marginal Cost C'(x)

P'(x), R'(x), and C'(x) are the instantaneous rates of change of profit, revenue, and cost respectively.

Example 1

The marginal profit of manufacturing and selling a certain drug is P'(x )=100 – 0.005x.

How much profit should the company expect if it sells 10,000 units of this drug?

Example 2

If the marginal cost of producing x units of a commodity is C'(x) = 5 + 0.4x, Find

1. the marginal cost when x =50;
2. the cost of producing the first 100 units.

Solution:

1. Marginal cost at x = 50:

C'(50) = 5 + 0.4(50) = 5 + 20 = 25.

2. Cost of producing 100 units:

Temperature Problems

Example

On a certain day, the changes in the temperature in a greenhouse beginning at 12 noon are represented by f (t)= sin degrees Fahrenheit, where t is the number of hours elapsed after 12 noon. If at 12 noon, the temperature is 95°F, find the temperature in the greenhouse at 5 p.m.

Let F(t) represent the temperature of the greenhouse.

The temperature in the greenhouse at 5 p.m. is 98.602 °F.

Leakage Problems

Example

Water is leaking from a faucet at the rate of l(t) =10e^–0.5t gallons per hour, where t is measured in hours. How many gallons of water will have leaked from the faucet after a 24 hour period?

Let L(x) represent the number of gallons that have leaked after x hours.

Using your calculator, enter (10e^{^}(–0.5x), x, 0, 24) and obtain 19.9999. Thus, the number of gallons of water that have leaked after x hours is approximately 20 gallons.

Growth Problem

Example

In a farm, the animal population is increasing at a rate which can be approximately represented by g(t) = 20 + 50 ln(2 + t), where t is measured in years. How much will the animal population increase to the nearest tens between the 3rd and 5th year?

Let G(x) be the increase in animal population after x years.

Thus, the population increase between the 3rd and 5th years

Enter (20 + 50 ln(2 + x), x, 3, 5) and obtain 218.709.

Thus the animal population will increase by approximately 220 between the 3rd and 5th years.

Practice problems for these concepts can be found at: Applications of Definite Integrals Practice Problems for AP Calculus