Area by Upper and Lower Sum
The main goal of using the upper and lower sum is to approximate the area by using rectangles. The sum is calculated by dividing the region up into shapes (rectangles, trapezoids, etc). To compute the area under a curve we will use rectangles, and we make approximations by using rectangles inscribed in the curve and circumscribed on the curve. The total area of the inscribed rectangles is the lower sum, and the total area of the circumscribed rectangles is the upper sum. By taking more rectangles, we will get a better approximation. in upper sums, the rectangles height is determined by where a corner of the box hits the curve, and the rectangles are draw above the curve as shown in the graph. In lower sums, the rectangles height is determined by where a corner of the box hits the curve, and the rectangles are draw below the curve as show.
I used Geogebra to find the lower and upper sum for the function f(x) = 1/4 (x^3-4). I have four sliders, “a”, “b”, “n”, and “p”. The “p” slider is for rectangle sums in order to see the difference between the lower, upper, and rectangle sums. Slider “p” gives the position where to start drawing the rectangle. Slider “a” gives the start x-value, slider “b” gives the end x-value, and slider “n” gives the number of rectangles we use to do the approximation.
The command that I used are:
• uppersum[f,a,b,n], with green color on geogebra graph
• lower sum[f,a,b,n], with red color on geogebra graph
• rectanglesum[f,a,b,n,p], with blue color on geogebra graph
If you noticed when you have the “p” slider start at p = 0 the rectangle sum and the upper sum will have the same estimate. If p = 1 the rectangle sum and the lower sum will have the same estimate