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DECK AREA REVIEW

Math Topic: Riemann Sums
Key Terms: approximations, endpoints, summation notation

The Math behind the Observations:

Riemann Sums and Area:

The diagram above shows a static version of the Deck Area applet. In the lab exercise the first question refers to this scene. We note that using the left endpoints the error is greatest on the right where the curve is decreasing. Thus, with 2 rectangles the approximation is bigger than the actual area.

This observation can be extended to a more complete discussion of Riemann Sums and approximating areas. In the following we present clips from a tutorial calculus series from Harvey Mudd College together with observations about the Riemann Sum process.

Clip 1. Problem Posed - Finding the Area under a Curve:

In the clip below the original problem is presented: Finding the area of the region R bounded above by the curve y = f(x), below by the x-axis, and on the sides by the lines x=a and x=b. Throughout this discussion, f will be a continuous function.

Clip 2. Partitioning [a,b]:

In the this clip we take the first step in the Riemann process, dividing [a,b] into subintervals, [x₀,x₁],[x₁,x₂],…[x_n-1,x_n], where a=x₀ < x₁ <… < x_n = b. This is the partition of the interval. The lengths of the subintervals ( which need not be the same) are D x₁, D x₂, ….,D x_n, respectively.

Clips 3 & 4. Approximating each strip with a rectangle:

At this step each strip is to be approximated by a rectangle with height equal to the height of the curve y= f(x) at some arbitrary point in the subinterval. Thus, for the first subinterval [x₀,x₁], choose some x₁^* within the interval and use f(x₁^*) as the height of the first rectangle.

Note: The area of the corresponding rectangle will be the height times the base, and so equals f(x₁^*)D x₁.

Clip 3 shows this first step.

In the next clip, Clip 4, we repeat the process of forming rectangles for each subinterval.

Note: In the above clip we see that, depending on what points we choose for the x_i^*, our approximations may be too large or too small.

Clip 5. Overestimates and Upper Sums:

In the clip below each x_i^* has been chosen to be a point in the subinterval giving maximum height. Thus, when we add up the rectangles we will have an overestimate of the area of R. This sum is called an upper sum.

Clip 6. Underestimates and Lower Sums:

In this next clip we choose each x_i^* to be the point giving minimum height. Thus, summing up the rectangles yields an underestimate to the real area of R. We call this type of sum a lower sum.

Clip 7. Random Points and Riemann Sums:

In this clip points are chosen randomly. In this case the sum, å _iⁿ₌₁ f(x_i^*) D x_i

is called a Riemann Sum. Note that the Riemann Sum also gives an approximation to the area of R that is between the lower and upper sums. The upper and lower sums can also be regarded as specific Riemann sums.

Clip 8. Riemann Sums and the Definite Integral:

In the final clip we take the limit of all sums as the number of rectangles goes to infinity and the widths of the subintervals go to zero. This limit, if it exists, is the definition of the definite integral, , and this provides the true area of the region R.

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USNA Mathematics Department
Comments to: Professor Carol G. Crawford, at cgc@nadn.navy.mil or Professor Mark D. Meyerson, at mdm@nadn.navy.mil