Riemann Sums

1. a. A car accelerates from a standing start at (approximately) the rate of

,

where *t* is the number of seconds since starting. Use Graphing Calculator and Riemann sums to produce a graph of the approximate velocity (in mi/hr) of the car at each moment during its first 10 seconds of accelerating.

Pick a point on the graph of your function. What are its coordinates? What do its coordinates represent? What does the graph represent?

b. Use Graphing Calculator and Riemann sums to produce a graph of the approximate distance covered (in miles) at each moment during a car's first 10 seconds of accelerating from a standing start when it accelerates at the rate presented in part (a). Use the method that the interval (0,x) is partitioned into *n* subintervals.

c. Repeat part (b), but use the method that every interval (0,x) is partitioned into subintervals of length *d*.

2. a. Explain how to think of the expression

(where *p* is a fixed number of subintervals) to understand that it defines a function evaluated at *every* value of *x* in (*a*,*b*)? (Click *here* for a GC file giving a general definition.)

b. Do functions defined like this

(where *d* is an arbitrarily small interval size) *always* produce a step function? If so, why? How does this function differ from the one in (a)? (Click *here* for a GC file containing a general definition.)

3. Use Graphing Calculator and Riemann sums to produce a graph of the volume of water in a conical storage tank that is 25ft high and 30 feet wide at the top. Express the volume as a function of the height of the water above the tip of the cone.

4. Hexane is a gas used for industrial purposes. Clentice Smith of Cargill Corp., Bloomington, IL requested a graph that will give the approximate volume of hexane (measured in cubic inches) held by the tank shown in Figure 1.^{*} Use Graphing Calculator and Riemann sums to produce such a graph. Express the volume of hexane as a function of the height of the water (measured in inches). A diagram is given on the next page.

__Assumptions__

• The face of the tank is a disk (i.e., a region bounded by a circle)

• the shape of the tank is cylindrical

• the hexane sits atop the water

• the dimensions of the tank are as shown

• a hole in the tank resides 18" vertically from the top of the tank

• the hexane always reaches the bottom edge of the hole.

Figure 1

5. a. Find two applications of the integral in a calculus text, *not* having to do with area or volume, for which these methods are appropriate. Solve them using these methods. (*Beware, the answer the textbook seeks will probably be a number instead of a function. So, you will give a more general answer than they request, but you can still answer their question*.)

b. Discuss how the *question* might need to be changed and how the *situation* might need to be re-framed to make Riemann sums a reasonable method.

* Mr. Smith actually requested a table of values so that he could put a ruler along the face of the tank and read the volume of hexane from the table by reading the height of the water on the ruler.