Suppose is a non-negative, continuous function defined on some interval . Then by the area under the curve between and we mean the area of the region bounded above by the graph of , below by the -axis, on the left by the vertical line , and on the right by the vertical line . All of the numerical methods in this lab depend on subdividing the interval into subintervals of uniform length.

In these simple rectangular approximation methods, the area above each subinterval is approximated by the area of a rectangle, with the height of the rectangle being chosen according to some rule. In particular, we will consider the left, right and midpoint rules.

The Maple `student` package has commands for visualizing these
three rectangular area approximations. To use them, you first must
load the package via the with command. Then try the three commands
given below to help you understand the differences between the
three different rectangular approximations. Note that
the different rules choose rectangles which in
each case will either underestimate or overestimate the area.

> with(student): > rightbox(x^2,x=0..4,3); > leftbox(x^2,x=0..4,16); > middlebox(x^2,x=0..4,10);There are also Maple commands

> rightsum(x^2,x=0..4); > evalf(rightsum(x^2,x=0..4,3)); > evalf(leftsum(x^2,x=0..4,16)); > evalf(middlesum(x^2,x=0..4,10));

All of the Maple commands described so far in this lab can include a third
argument to specify the number of subintervals. The default is 4
subintervals. The example below approximates the area under
from to using the `rightsum` command with 50,
100, 320 and 321 subintervals. As the number of subintervals
increases, the approximation gets closer and closer to the exact
answer. You can see this by assigning a label to the approximation,
assigning a label to the exact answer and taking their
difference. The closer you are to the actual answer, the smaller the
difference. The example below shows how we can use Maple to
approximate this area with an absolute error no greater than 0.1.

> exact := 4^3/3; > estimate := evalf(rightsum(x^2,x=0..4,50)); > evalf(abs(exact-estimate)); > estimate := evalf(rightsum(x^2,x=0..4,100)); > evalf(abs(exact-estimate)); > estimate := evalf(rightsum(x^2,x=0..4,320)); > evalf(abs(exact-estimate)); > estimate := evalf(rightsum(x^2,x=0..4,321)); > evalf(abs(exact-estimate));

- Given the function
- Plot on the interval with 8 rectangles determined by the right-endpoint rule. Then plot the function again with 8 rectangles using the left-endpoint rule.
- Which rule overestimates the area for the given function? Will this rule always overestimate the area for any function? Why?
- Is the midpoint rule an average of the left and right endpoint rules? Verify your claim using the above function with 6 rectangles.

- The area under the graph of , above the -axis, over the interval represents the area of half of a circle of radius 1. So, the exact area would be . Find the minimum number of rectangles needed using the left endpoint rule to approximate this area with absolute error no greater than 0.01. Repeat this using the midpoint rule. Which method is better and why.

2011-03-18