Problem 2 How much water is flowing in a river? The U.S. Geological Survey (USGS) operates more than 8200 stations tracking the streamflow of rivers, streams, and creeks across the U.S. One of the properties they monitor is discharge, or the volume of water flowing through a cross-sectional area of a channel per unit time, measured in units of cubic feet per second (cfs). The USGS uses a Riemann sum to estimate discharge, as shown in the diagram on the right. The cross- sectional area is divided up into subsections of a certain width. Each subsection is approximated by a rectan- gle, where the height of the rectangle is determined by measuring the depth of the water at a certain point within the subsection. The velocity of the water is also measured at this point. The discharge for the subsec- il Area = Dapth x Width tion is then approximated by the area of the rectangle Dept Discharge = Area x Valocity times the measured velocity. Summing these gives the . ) ) . : : ' For more details on how discharge is measured, sece estimated dlbthdl'ge for the channel. the overview on the USGS website. Suppose that the discharge for the Potomac River is to be measured at a certain point where the surface width of the river is 2000 ft. Define the following: e is the displacement (in feet) from the center of the river, where positive values are towards the Maryland/D.C. shoreline and negative values are towards the Virginia shoreline; e h(x) is the depth (in feet) of the water x feet from the center; e v(x) is the velocity (in feet per second) of the water z feet from the center. Assume that: e this cross-section of the Potomac is symmetric, so hi(a) and v(x) are even functions, e h(x) and v(z) are decreasing on 0