Getting the Most Out of the
Fundamental Theorem
y=f(t)
a
x
b
Dan Kennedy
Baylor School
Chattanooga, TN
dkennedy@baylorschool.org
One of the hardest calculus topics to
teach in the old days was Riemann
sums.
They were hard to draw, hard to
compute, and (many felt) totally
unnecessary.
That was why most of us quickly
moved on to antiderivatives, which
is how we wanted students to do
integrals.
Needless to say, when we came to
the Fundamental Theorem,
students found it to be the greatest
anticlimax in the course.
Integration and differentiation are
reverse operations? Well, duh.
Then along came the TI graphing
calculators. Using the integral utility in
the CALC menu, students could
actually see an integral accumulating
value from left to right along the x-axis,
just as a limit of Riemann sums would
do:
So now we can do all kinds of summing
problems before we even mention an
antiderivative.
Historically, that’s what scientists had
to do before calculus.
Here’s why it mattered to them:
mi/hr
60
v(t) = 40
40
d = 120 mi
20
1
4
hr
The calculus pioneers knew that the
area would still yield distance, but what
was the connection to tangent lines?
And was there an easy way to find these
irregularly-shaped areas?
mi/hr
60
v(t)
40
20
d = 120 mi
1
4
hr
Since the time of Archimedes,
scientists had been finding areas of
irregularly-shaped regions by dividing
them into regularly-shaped regions.
That is what Riemann sums are all
about.
2.033281
2.008248
2.000329
With graphing calculators, students can
find these sums without the tedium.
They can also imagine the tedium of
doing these sums by hand!
Best of all, they can actually see the
limiting case:
And the calculator shows the thin
rectangles accumulating from left to
right – ideal for understanding the FTC!
Let us consider a positive continuous
function f defined on [a, b].
Choose an arbitrary x in [a, b].
y=f(t)
a
x
b
Each choice of x determines a unique
area from a to x, denoted as usual by

x
a
f (t )dt
y=f(t)
a
x
b
Thus

x
a
f (t )dt
is a function of x on [a, b].
What is the derivative of this function?
xh

d x
a
f (t )dt  lim

h 0
dx a

 lim
h 0
xh
x
x
f (t )dt   f (t )dt
a
h
f (t )dt
h
xh

d x
a
f (t )dt  lim

h 0
dx a

 lim
h 0
xh
x
x
f (t )dt   f (t )dt
a
h
f (t )dt
h
 f ( x)
y=f(t)
a
x x+h
b
So
d x
f (t )dt  f ( x).

dx a
But that is only half the story.

x
Now that we know that
f (t )dt
a
an antiderivative of f,
we know that it differs from any
antiderivative of f by a constant.
is
That is, if F is any antiderivative of f,

x
a
f (t )dt  F ( x)  C.
To find C, we can plug in a:

a
a
f (t )dt  F (a )  C
0  F (a)  C
C   F (a )
So

x
a
f (t )dt  F ( x)  F (a ).
Now plug in b:

b
a
f (t )dt  F (b)  F (a).
This was the FTC. This was the result
that changed the world.
2.000329
Now, instead of wasting a full afternoon just to
get an approximation of the area under one arch
of the sine curve, you could find one
antiderivative, plug in two numbers, and subtract!


0
sin( x)dx   cos( x)

0
  cos( )  cos(0)  2.
Since 2000, the AP Calculus Test
Development Committee has been
emphasizing a conceptual understanding
of the definite integral, resulting in these
“new” problem types:
Functions defined as integrals
Accumulation Problems
Integrals from Tables
Finding f (b) , given f ( a ) and f ( x)
Interpreting the Definite Integral
Problem of the Day #29:
Suppose

x
1
f (t )dt  x 3  2 x  k .
(a) Find f(x).
(b) Find k.
(a) By the Fundamental Theorem,
d x
d 3
f
(
t
)
dt

x

2
x

k
)



dx 1
dx
2
f ( x)  3x  2
(b) Plug in x = 1:

1
1
f (t )dt  1  2(1)  k
3
0  1  k
k 1
Quick Aside:
My First Day
on the AP
Calculus Test
Development
Committee
Here was the problem (1987):
If f ( x)  x2 , which of the following could be the graph of
x
y   f (t )dt ?
1
(A)
(B)
(D)
(E)
(C)
This problem had been checked:
1. by the author who had written it;
2. by the committee that had okayed it;
3. by the committee that had okayed it for a
pre-test;
4. by the ETS test development specialists;
5. by our committee, reviewing the final form of
the college pre-test.
My colleagues were two problems further into
the test when I asked if we could go back for
another look.
The proposed key was (B). That is,
x
If f ( x)  x2 , the graph of y  1 f (t )dt could be
(B)
While everyone was concentrating on the
Fundamental Theorem application, they had
missed the hidden “initial condition” that y
must equal zero when x = 1!
Here’s 1995 / BC-6:
2
1
0
-1
-2
-3
1
2
3
4
5
Let f be a function whose domain is the closed interval [0, 5].
The graph of f is shown above.
Let h( x)  
(a)
(b)
(c)
x
3
2
0
f (t )dt.
Find the domain of h.
Find h(2).
At what x is h(x) a minimum? Show the analysis that
leads to your conclusion.
(a) The domain of h is all x for which

x
3
2
0
f (t )dt is defined:
x
0 35  6 x  4
2
(b) A little Chain Rule:
x
 1
f (t )dt  f   3  
2
 2
1
3
h(2)  f (4)   
2
2
d
h( x)  
dx
x
3
2
0
2
1
0
-1
-2
-3
1
2
3
4
5
Let f be a function whose domain is the closed interval [0, 5].
The graph of f is shown above.
Let h( x)  
(a)
(b)
(c)
x
3
2
0
f (t )dt.
Find the domain of h.
Find h(2).
At what x is h(x) a minimum? Show the analysis that
leads to your conclusion.
(c) Since

x
3
2
0
f (t )dt is positive from
-6 to -1 and negative from -1 to 4,
the minimum occurs at an endpoint.
By comparing areas, h(4) < h(-6) = 0,
so the minimum occurs at x = 4.
This “area comparison” genre of
problem was pretty common in the
early graphing calculator days.
Accumulation Problems
Perhaps the most groundbreaking change
in the 1998 AP Course Description was
the decision not to list the “applications
of integration” that a student should
know. Instead, students would be
expected to have enough familiarity with
“accumulation” problems to model them
with integrals in fresh situations.
The exams since then have provided an
abundance of fresh situations!
All students should know how to
interpret the following applications as
accumulations:
Areas (sums of rectangles)
Volumes (sums of regular-shaped slices)
Displacements (sums of v(t)∙∆t)
Average values (Integrals/intervals)
BC: Arclengths (sums of hypotenuses)
BC: Polar areas (sums of sectors)
Problem of the Day #27:
Find the average length of all chords of a circle of
radius r that are perpendicular to a given diameter.
I give this before the FTC and before
any definition of average value.
Answer: Take the area of the circle and
divide it by the diameter!
Average chord length =
r
2
2r

r
2
.
Problem of the Day #45:
The population density of the city of Washerton decreases as you move
away from the city center. In fact, it can be approximated (in people per
square mile) by the function 10,000(2 – r) at a distance r miles from the
city center.
(a) What is the radius of the populated portion of the city?
(b) A thin ring around the center of the city has thickness r and
radius r. What is its area? [Hint: Imagine straightening it out to
make a thin rectangular strip.]
(c) What is the population of the strip in part (b)?
(d) Estimate the total population of Washerton by setting up and
evaluating a definite integral.
The population density of the city of Washerton decreases as you move
away from the city center. In fact, it can be approximated (in people per
square mile) by the function 10,000(2 – r) at a distance r miles from the
city center.
(a)
What is the radius of the populated portion of the city?
(a) r = 2 miles.
(b) A thin ring around the center of the city has thickness r and
radius r. What is its area? [Hint: Imagine straightening it out to
make a thin rectangular strip.]
(b) A = 2πr Δr
(d) Estimate the total population of Washerton by setting up and
evaluating a definite integral.
(d)

2
0
10,000(2  r )(2 r )dr
people per sq. mile
 83,776 people
sq. miles
Problem of the Day #49:
What is the length of one complete "S" of the sine curve? (That
is, how long would it be if stretched out?) Work with your partner
to estimate it without consulting the textbook, but then the two of
you will need to make distinct guesses. (It is an irrational number,
so the probability of estimating it exactly is zero.) Best estimate
wins a prize.
y


x










2500
2007 / AB-2 BC-2
2000
1500
1000
500
1
2
3
4
5
6
7
The amount of water in a storage tank, in gallons, is modeled by a
continuous function on the time interval 0  t  7 , where t is measured in
hours. In this model, rates are given as follows:
(i)
The rate at which water enters the tank is f (t )  100t 2 sin t
 
gallons per hour for 0  t  7 .
(ii) The rate at which water leaves the tank is
250 for 0  t  3
g (t )  
gallons per hour.
2000 for 3  t  7
The graphs of f and g, which intersect at t = 1.617 and t = 5.076, are
shwn in the figure above. At time t = 0, the amount of water in the tank
is 5000 gallons.
2500
2000
1500
1000
500
1
2
3
4
5
6
7
(a) How many gallons of water enter the tank during the time interval
0  t  7 ? Round your answer to the nearest gallon.
(b) For 0  t  7 , find the time intervals during which the amount of
water in the tank is decreasing. Give a reason for each answer.
(c) For 0  t  7 , at what time t is the amount of water in the tank
greatest? To the nearest gallon, compute the amount of water at this
time. Justify your answer.
2500
2000
1500
1000
500
1
2
3
4
5
6
7
Problem of the Day # 30
We measure the speed of a bobsled at one-second intervals as it begins its
run:
Seconds
Speed (mph)
0
0
1
6
2
12
3
25
4
40
5
55
6
70
7
80
8
90
9
95
10
100
(a) What is the bobsled's approximate acceleration when t = 4 seconds?
(b) About how far does the bobsled travel in the first 10 seconds?
(Caution: Do not use regression on your calculator.)
2001 / AB-2 BC-2
t
W(t)
(days)
(  C)
0
20
The table to the right shows the water temperatures as
3
31
recorded every 3 days over a 15-day period.
6
28
9
24
(a) Use data from the table to find an approximation for
12
22
15
21
W (12) .
Show the computations that lead to your answer. Indicate units of measure.
(b) Approximate the average temperature, in degrees Celsius, of the water over the time
interval 0  t  15 days by using a trapezoidal approximation with subintervals of
length t = 3 days.
(c) A student proposes the function P, given by P(t )  20  10te(  t / 3) , as a model for the
temperature of the water in the pond at time t, where t is measured in days and P(t) is
measured in degrees Celsius. Find P(12) . Using appropriate units, explain the
meaning of your answer in terms of water temperature.
(d) Use the function P defined in part (c) to find the average value, in degrees Celsius, of
P(t) over the time interval 0  t  15 days.
The temperature, in degrees Celsius (C), of the water in a
pond is a differentiable function W of time t.
Problem of the Day #35:
If F ( x)  sin ( x) and F(2) = 5, find F(7).
2
Another implication of the Fundamental
Theorem (and a source of several
recent problems that have caused
trouble for students):

b
a
f ( x)dx  f (b)  f (a )
b
 f (b)  f (a )   f ( x)dx
a
Thus, given f(a) and the rate of change
of f on [a, b], you can find f(b).
The Kicker in 2003 / AB-4 BC-4:
y
2
(–3, 1)
–4
–2
2
4
x
–2
(4, –2)
Let f be a function defined on the closed interval 3  x  4 with f(0) = 3.
The graph of f  , the derivative of f, consists of one line segment and a
semicircle, as shown above.
(d) Find f(–3) and f(4). Show the work that leads to your answers.
f (3)  f (0)  
3
0
f ( x) dx
1
1
 3  (2)(2)  (1)(1)
2
2
9

2
4
f (4)  f (0)   f ( x) dx
0
 3  (8)  (2 )
 2  5
dkennedy@baylorschool.org