Riemann+Sum

**__Riemann Sums__** Riemann sums are used to solve integrals. They find the area underneath a curve using three different forms of riemann sums, which are: left-hand, right-hand, and midpoint. Midpoint Riemann sums are the most accurate of the three. To solve an integral using Riemann sums, a set sub-interval is used to create rectangles of the same width and then finding the area within the rectangles is the value of the area under the curve. For a left-hand sum, the formula is: On a graph, using the left-hand Riemann sum, a line is drawn from the beginning (left side) of the set interval up to the line of the graph and then right to the end of the set interval. Left-hand Riemann sums underestimate the area under the graph.

In this example the set interval is 0.5, and proves that left-hand Riemann sums underestimate the area because there is still space between the rectangles and the line of the curve. For a right-hand sum, the formula is: On a graph, using the right-hand Riemann sum, a line is drawn from the end (right side) of the set interval up to the line of the graph and then left to the set interval. Right-hand Riemann sums over-estimate the area under the curve.

In this example the set interval is 0.5, and proves that the right-hand Riemann sum over-estimates the area because there is more area from the rectangles above the curve.

For a midpoint sum, the formula is:

On a graph, using the mid-point Riemann sum, a line is drawn from the middle of the set interval up to the line of the graph and then both right and left to span the set interval. Mid-point Riemann sums are the most accurate of the three forms of Riemann sums, and it could underestimate as well over estimate depending on the line of the equation.



In this example the set interval is 0.5, and the sum above is an over estimate because the area in the rectangles is slightly larger than the actual area under the curve.

__Problem 1:__


 * t(hours) || 0 || .5 || 1 || 1.5 || 2 || 2.5 || 3 ||
 * v(miles/hour) || 32 || 30 || 16 || 22 || 20 || 24 || 26 ||

A speedboat travels downstream on a river. Its speed v, in miles per hour, at certain times is given in the table above. Using a left Riemann sum, what is the approximation of the total distance traveled by the speedboat from t = .5 to t = 3?

A) 52 miles B) 56 miles C) 46 miles D) 66 miles
 * To view answer go on edit and highlight the space under the sumation equation with the cursor.**

Answer: d = 30(1-.5) + 16(1.5-1) + 22(2-1.5) + 20(2.5-2) + 24(3-2.5). d = 56 miles

__ Problem 2: __ The values of a differentiable function f are given in the table below. Aproximate the integral of this function from x = 0 to x = 8, by using a right Riemann sum.

A) 50 B) 42 C) -52 D) 52
 * x || 0 || 2 || 4 || 6 || 8 ||
 * f(x) || -10 || -2 || 6 || 8 || 14 ||
 * To view the answer go on edit and highlight the space under "Answer:" with the cursor.**

Answer: R = -2(2-0) + 6(4-2) + 8(6-4) + 14(8-6). R = 52