§ 6.5 Applications of the Definite Integral

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

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

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

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

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

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

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

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

Future Value of an Income Stream EXAMPLE (Future Value) Suppose that money is deposited daily into a savings account at an annual rate of $2000. 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

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 0 ≤ 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

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

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