MATH 163, Spring 2001 Due Date: Name(s):
Project 5.3a: The Fundamental Theorem of Calculus
Objective
To illustrate the Fundamental Theorem of Calculus.
Narrative
If you have not already done so, read Section 5.3 of the text. One of the key ideas you should come away
from this section with is the Fundamental Theorem of Calculus. This result states that if f is a continuous
function on the finite closed interval [a, b] then
Dx
x
t=a
f(t) dt = f(x)
for x  (a, b), and hence
b
x=a
f(x) dx = F(b) - F(a)
where F is any antiderviative for f (that is, F (x) = f(x)).
Task
a) Type the command lines in the left-hand column below into Maple in the order in which they are
listed. These commands are aimed at investigating the Fundamental Theorem of Calculus in the context of
4
x=1
4/(x2
+ 1) dx. The effect of each command is described in the right-hand column for your reference.
Your lab report will be a hard copy of your typed input and Maple's responses (both text and graphics).
> # Project 5.3a: The Fundamental Theorem of Calculus
> restart; Clear Maple's memory.
> f := x -> 4/(x^2+1); Let f(x) = 4/(x2
+ 1).
> plot(f(x),x=-5..5,scaling=constrained); Plot the graph of f.
> a := 1.0; b := 4.0; Let a = 1 and b = 4.
> limit(sum(f(a+i*(b-a)/n)*(b-a)/n,i=1..n),n=infinity); Find
b
x=a
f(x) dx thinking of it as
an area.
> F := x -> int(f(t),t=1..x); Let F(x) =
x
t=1
f(t) dt. (We're
defining F as an integral.)
> plot({f(x),F(x)},x=1..4); Plot the graphs of f and F from
x = 1 to x = 4.
> D(F)(x); The first part of the Fundamental
Theorem of Calculus states that
the derivative of F is f.
> F(b) - F(a); Find F(b) - F(a).
At this point, make a hard-copy of your typed input and Maple's responses. Then proceed with the following
steps.
b) Label the graphs of f and F in the plot you made in part (a) by hand.
c) Plot and label the point P(3, F(3)).
d) The value F(3) represents an area. Shade in the area that corresponds to F(3). (Be careful!)