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); > leftbox(x^2,x=0..4); > middlebox(x^2,x=0..4);There are also Maple commands

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

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));

- Consider the function
on the interval .
- Use the command
`rightsum`to approximate area above the -axis and under with 10, 25 and 50 rectangles. - The exact value of the area under over the interval is
. Enter this value into Maple and label it
`exact`. Look at the decimal version of the exact answer and compare it to the answers above. Then use the command`rightbox`to explain why the`rightsum`command yields a value that is always less than the exact value of the area. You may use a value of in this plot. - Does the midpoint rule yield an approximation equal to an average of left endpoint and right endpoint rules? Verify for .
- Use the command
`rightsum`to find the minimum number of rectangles that approximates this area with absolute error no greater than 0.1.

- Use the command
- What is the area of the circle ? Solving this equation for as a function of and using the positive function represents the upper half of the circle whose area you are trying to approximate. Enter this function into Maple and the exact area as half the area of a circle of radius 4:
f:=x->sqrt(16-x^2); exact:=16*Pi/2;

Use the area approximations`leftsum`and`middlesum`to determine the minimum number of rectangles required to estimate the area of the circle with error no greater than 0.1. Based on your results, which approximation method is better, left hand endpoint rule or the midpoint rule?

2013-01-13