A Riemann Sum begins with the question of how to find the area under a curve (i.e., between a positive curve and the x-axis, which is essentially a Geometry question). It turns out, though, that as with many seemingly limited questions in mathematics, we can expand the application of the answer to this question to other areas (e.g., finding distance travelled by a moving object in physics). In order to do this, we must generalize the initial question using what is called a Riemann Sum.

We begin with finding the area under a positive curve (i.e., a curve with positive y-values). For example, suppose we wish to find the area under the curve ##f(x) = x^2## from ##x = 1## to ##x = 3##. We divide this interval into “subintervals.” For example, we might choose four subintervals of equal width (1-1.5, 1.5-2, 2-2.5, 2.5-3).

We then approximate the area of each subinterval using a rectangle. The width of the rectangle is the the change in x which is 0.5 (or ##Delta x = 0.5##). The height is a selected y-value for each subinterval. In this case, we shall choose the right-hand endpoints. Thus we have the following rectangular areas:

##f(1.5)*0.5+f(2)*0.5+f(2.5)*0.5+f(3)*0.5 = 10.75##.

Note that we can abbreviate this using summation (sigma) notation as:

##sum_(i=1)^4(f(1+i*0.5)*(0.5))## or more generally as ##sum_(i=1)^4(f(1+i*Deltax)*(Deltax))##.

We now finally come to the idea of a Riemann Sum. A Riemann Sum is encountered when we relax our rules for situations like the previous one to allow for a more general view of the problem. That is, we could use different values for each ##Delta x## and select various points for each ##f(x)##. We could also include functions that are not always positive.

The Riemann Sum would then look like this:
##sum_(i=1)^n(f(barx_i)*(Deltax_i))##, where ##barx_i## is some x value in the ith. subinterval and ##Deltax_k## is the width of that subinterval.

Some of our products might now be negative, if some of the y-values are negative. These would represent “negative areas” (i.e., areas below the x-axis). The Riemann Sum would be an approximation of the difference between the “positive” and “negative” areas.

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