Calculus AB and Calculus BC

CHAPTER 8 Further Applications of Integration

C. OTHER APPLICATIONS OF RIEMANN SUMS

We will continue to set up Riemann sums to calculate a variety of quantities using definite integrals. In many of these examples, we will partition into n equal subintervals a given interval (or region or ring or solid or the like), approximate the quantity over each small subinterval (and assume it is constant there), then add up all these small quantities. Finally, as n → ∞ we will replace the sum by its equivalent definite integral to calculate the desired quantity.

EXAMPLE 8

Amount of Leaking Water. Water is draining from a cylindrical pipe of radius 2 inches. At t seconds the water is flowing out with velocity v(t) inches per second. Express the amount of water that has drained from the pipe in the first 3 minutes as a definite integral in terms of v(t).

SOLUTION: We first express 3 min as 180 sec. We then partition [0,180] into n subintervals each of length Δt. In Δt sec, approximately v(t) Δt in. of water have drained from the pipe. Since a typical cross section has area 4π in.2 (Figure N8–2), in Δt sec the amount that has drained is

(4π in.2) (v(t) in./sec)(Δt sec) = 4πv(t) Δt in.3.

The sum of the n amounts of water that drain from the pipe, as n → ∞, is Image the units are cubic inches (in.3).

Image

FIGURE N8–2

EXAMPLE 9

Traffic: Total Number of Cars. The density of cars (the number of cars per mile) on 10 miles of the highway approaching Disney World is equal approximately to f (x) = 200[4 − ln (2x + 3)], where x is the distance in miles from the Disney World entrance. Find the total number of cars on this 10-mile stretch.

SOLUTION: Partition the interval [0, 10] into n equal subintervals each of width Δx. In each subinterval the number of cars equals approximately the density of cars f (x) times Δx, where f (x) = 200[4 − ln (2x + 3)]. When we add n of these products we get Image which is a Riemann sum. As n → ∞ (or as Δx → 0), the Riemann sum approaches the definite integral

Image

which, using our calculator, is approximately equal to 3118 cars.

EXAMPLE 10

Resource Depletion. In 2000 the yearly world petroleum consumption was about 77 billion barrels and the yearly exponential rate of increase in use was 2%. How many years after 2000 are the world’s total estimated oil reserves of 1020 billion barrels likely to last?

SOLUTION: Given the yearly consumption in 2000 and the projected exponential rate of increase in consumption, the anticipated consumption during the Δtth part of a year (after 2000) is 77e0.02t Δt billion barrels. The total to be used during the following N

years is therefore Image This integral must equal 1020 billion barrels.

We must now solve this equation for N. We get

Image

Either more oil (or alternative sources of energy) must be found, or the world consumption must be sharply reduced.