The below graph, s(t), represents the rate at which snow falls, in centimeters per hour, over the six hour period from 12 a.m. to 6 a.m.

(a) Design an equation, f(t), which represents the total snowfall since 12 a.m. at any time t.

(b) Approximate the value of f(3) using upper Riemann sums with 6 rectangles.

(c) Approximate f(6) using midpoint Riemann sums with 6 rectangles.

You may use a graphing calculator for this activity · Difficulty:

Solution

(a) The area beneath a rate curve represents total change. For example, the area beneath a velocity curve over a set period of time represents the distance traveled over that time. Therefore, the definite integral bounded below by 0 and above by the time in question will give you the total snowfall at that moment: .

(b) Upper sums are found by splitting the x-axis into 6 parts (since the problem specific 6 rectangles). As we're approximating the area beneath s(t) on [0,3], each rectangle must be 1/2 unit wide. On each of the 6 half-unit wide intervals, you have to pick the point at which the curve reaches it's highest value, and use that as the height of the rectangle. In essence, you're using these rectangles:

We don't know exactly how tall those rectangles are, since we don't have a formula for s(t), so you have to eyeball it. (We're just approximating anyway.) The area of each rectangle is length times width:

You may not have chosen the same approximations as I did, so your answer may vary slightly. Since this is an upper sum, it's value is larger than the actual snowfall.

(c) In this approximation, each rectangle will have a width of 1, and its height will be defined by how high the function s(t) is in the exact middle of each interval. For example, the rectangle spanning from 3 to 4 will have the height s(3.5). Again, we'll have to approximate the function values with our eyeballs: