Figure N8− 1C. 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