1. § 6.5
Applications of the Definite Integral

2. Section Outline Average Value of a Function Over an Interval
Consumers’ Surplus
Future Value of an Income Stream
Volume of a Solid of Revolution

3. Average Value of a Function Over an Interval
Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #55

4. Average Value of a Function Over an Interval
EXAMPLE
Determine the average value of f (x) = 1 – x over the interval -1 ≤ x ≤ 1.
SOLUTION
Using (2) with a = -1 and b = 1, the average value of f (x) = 1 – x over the interval -1 ≤ x ≤ 1 is equal to
An antiderivative of 1 – x is Therefore,
So, the average value of f (x) = 1 – x over the interval -1 ≤ x ≤ 1 is 1.
Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #56

5. Average Value of a Function Over an Interval
EXAMPLE
(Average Temperature) During a certain 12-hour period the temperature at time
t (measured in hours from the start of the period) was degrees. What was the average temperature during that period?
SOLUTION
The average temperature during the 12-hour period from t = 0 to t = 12 is
Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #57

6. Consumers’ Surplus
Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #58

7. Consumers’ Surplus
EXAMPLE
Find the consumers’ surplus for the following demand curve at the given sales level x.
SOLUTION
Since 20 units are sold, the price must be
Therefore, the consumers’ surplus is
Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #59

8. Consumers’ Surplus That is, the consumers’ surplus is $20. CONTINUED
Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #60

9. Future Value of an Income Stream
Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #61

10. Future Value of an Income Stream
EXAMPLE
(Future Value) Suppose that money is deposited daily into a savings account at an annual rate of $ If the account pays 6% interest compounded continuously, approximately how much will be in the account at the end of 2 years?
SOLUTION
Divide the time interval from 0 to 2 years into daily subintervals. Each
subinterval is then of duration years. Let t1, t2, ..., tn be points chosen
from these subintervals. Since we deposit money at an annual rate of $2000, the amount deposited during one of the subintervals is dollars. If this amount is deposited at time ti, the dollars will earn interest for the remaining 2 – ti years. The total amount resulting from this one deposit at time ti is then
Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #62

11. Future Value of an Income Stream
CONTINUED
Add the effects of the deposits at times t1, t2, ..., tn to arrive at the total balance in the account:
This is a Riemann sum for the function on the interval ≤ t ≤ 2. Since is very small when compared with the interval, the total amount in the account, A, is approximately
That is, the approximate balance in the account at the end of 2 years is $4250.
Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #63

12. Volume of a Solid of Revolution
Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #64

13. Volume of a Solid of Revolution
EXAMPLE
Find the volume of a solid of revolution generated by revolving about the x-axis the region under the following curve.
SOLUTION
Here g(x) = x2, and
Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #65