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. For example, dividing the interval [0,4] into four uniform pieces produces the subintervals , , , and .

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. When using the left endpoint rule, the height of the rectangle is the value of the function at the left-hand endpoint of the subinterval. When using the right endpoint rule, the height of the rectangle is the value of the function at the right-hand endpoint of the subinterval. The midpoint rule uses the value of the function at the midpoint of the subinterval for the height of the rectangle.

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. Make sure you understand the differences between the
three different rectangular approximations. Take a moment to see 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));

In the case of the rectangular approximations considered in this lab,
the way to improve the approximation is to increase the number of
subintervals in the partition. 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 4, 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 error no greater than 0.1.

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

- For the function
over the interval
, use the
`rightbox`,`leftbox`, and`middlebox`commands to plot the rectangular approximation of the area above the -axis and under with 15 rectangles. State in your opinion which graph gives the best approximation to the area and give a reason why. Be sure to comment on the shape of the graph in your reasoning. - Consider the semi-circle bounded above by
and
below by the axis for
. The exact area is, of
course, . Using the
`middlesum`command, approximate the area with 5, 10, and 15 subintervals. Compute the absolute error for your approximation with 15 subintervals. - You are familiar with the relation
for computing the distance
traveled over time by an object moving at a constant velocity. If
the velocity of the object is not constant, but varies with
time, then the net distance, , traveled over the interval
is given by the integral

where is the speed of the object as a function of time. Suppose that the velocity is given by . Use the`middlesum`rule with 15, 25, and 50 subintervals to approximate the net distance traveled for . The exact answer is to nine decimal places. Can you explain why a large number of subintervals are needed to get a good approximation? Graphing should help.

2002-01-11