Problem

The graph of a stream's rate of flow over a 1-hour period is the graph of a sine curve. Assuming the rate of flow is measured every 10 minutes, starting at time 0, use the method of accumulation to aproximate the total volume of flow over this hour, accurate to 3 decimal places.

Solution

First, examine the graph of hours vs. rate of flow. Units of time are in hours; units of rate of flow are in cubic meters per second.

The volume of flow that we're interested in is the area under this curve from time = 0 to time = 1. We approximate this volume assuming the rate of flow is constant over 10-minute intervals. The flow is measured at the beginning of each time interval. Using this assumption, all we need to do is calculate the area of each small recangle and add all areas together.

The width of each subinterval is

Since the rate of flow is given to us in cubic meters per second, it needs to be converted to cubic meters per hour for the calculations to make sense. Define the function f(t) to equal the flow rate per hour.

The area under the curve is therefore

Substitute for f(t) and simplify.