Contributed by:
In this section, we will learn that:
We get the same special type of limit in trying to find the area under a curve or a distance traveled.
2.
In Chapter 2, we used the tangent
and velocity problems to introduce
the derivative—the central idea in
differential calculus.
3.
In much the same way, this chapter starts
with the area and distance problems and
uses them to formulate the idea of
a definite integral—the basic concept of
integral calculus.
4.
In Chapters 6 and 8, we will see how to use
the integral to solve problems concerning:
Volumes
Lengths of curves
Population predictions
Cardiac output
Forces on a dam
Work
Consumer surplus
Baseball
5.
There is a connection between integral
calculus and differential calculus.
The Fundamental Theorem of Calculus (FTC)
relates the integral to the derivative.
We will see in this chapter that it greatly simplifies
the solution of many problems.
6.
5.1
Areas and Distances
In this section, we will learn that:
We get the same special type of limit in trying to find
the area under a curve or a distance traveled.
7.
AREA PROBLEM
We begin by attempting to solve
the area problem:
Find the area of the region S that lies
under the curve y = f(x) from a to b.
8.
AREA PROBLEM
This means that S,
illustrated here,
is bounded by:
The graph of a continuous
function f [where f(x) ≥ 0]
The vertical lines x = a and x
=b
The x-axis
9.
AREA PROBLEM
In trying to solve the area problem,
we have to ask ourselves:
What is the meaning of the word area?
10.
AREA PROBLEM
The question is easy to answer
for regions with straight sides.
11.
For a rectangle, the
area is defined as:
The product of the
length and the width
12.
The area of a
triangle is:
Half the base times the
height
13.
The area of a polygon
is found by:
Dividing it into triangles
and adding the areas
of the triangles
14.
AREA PROBLEM
However, it isn’t so easy to find the area
of a region with curved sides.
We all have an intuitive idea of what the area
of a region is.
Part of the area problem, though, is to make this
intuitive idea precise by giving an exact definition
of area.
15.
AREA PROBLEM
Recall that, in defining a tangent, we first
approximated the slope of the tangent line
by slopes of secant lines and then we took
the limit of these approximations.
We pursue a similar idea for areas.
16.
AREA PROBLEM
We first approximate the region S by
rectangles and then we take the limit of
the areas of these rectangles as we increase
the number of rectangles.
The following example illustrates the procedure.
17.
AREA PROBLEM Example 1
Use rectangles to
estimate the area under
the parabola y = x2 from
0 to 1, the parabolic
region S illustrated here.
18.
AREA PROBLEM Example 1
We first notice that the
area of S must be
somewhere between 0
and 1, because S
is contained in a square
with side length 1.
However, we can
certainly do better
than that.
19.
AREA PROBLEM Example 1
Suppose we divide S into
four strips
S1, S2, S3, and S4 by
drawing the vertical lines
x = ¼, x = ½, and x = ¾.
20.
AREA PROBLEM Example 1
We can approximate
each strip by a rectangle
whose base is the same
as the strip and whose
height is the same as the
right edge
of the strip.
21.
AREA PROBLEM Example 1
In other words, the
heights of these
rectangles are the values
of the function f(x) = x2
at the right endpoints of
the subintervals
[0, ¼],[¼, ½], [½, ¾],
and [¾, 1].
22.
AREA PROBLEM Example 1
Each rectangle has
width ¼ and
the heights are (¼)2,
(½)2, (¾)2, and 12.
23.
AREA PROBLEM Example 1
If we let R4 be the sum of the areas
of these approximating rectangles,
we get:
1 2 1 2 3 2 2
R4 1
4 4 1
4 2
1
4 4 1
1
4
15
32
0.46875
24.
AREA PROBLEM Example 1
We see the area A of
S is less than R4.
So, A < 0.46875
25.
AREA PROBLEM Example 1
Instead of using the
rectangles in this figure,
we could use the
smaller rectangles in
the next figure.
26.
AREA PROBLEM Example 1
Here, the heights are
the values of f at
the left endpoints of the
The leftmost rectangle
has collapsed because
its height is 0.
27.
AREA PROBLEM Example 1
The sum of the areas of these approximating
rectangles is:
2 1 2 1 2 3 2
L4 0
1
4
1
4 4 1
4 2 1
4 4
327
0.21875
28.
AREA PROBLEM Example 1
We see the area of S is
larger than L4.
So, we have lower and
upper estimates for A:
0.21875 < A < 0.46875
29.
AREA PROBLEM Example 1
We can repeat this
procedure with a larger
number of strips.
30.
AREA PROBLEM Example 1
The figure shows what
happens when
we divide the region S
into eight strips of equal
31.
AREA PROBLEM Example 1
By computing the sum of the areas of
the smaller rectangles (L8) and the sum of
the areas of the larger rectangles (R8),
we obtain better lower and upper estimates
for A:
0.2734375 < A < 0.3984375
32.
AREA PROBLEM Example 1
So, one possible answer to the
question is to say that:
The true area of S lies somewhere
between 0.2734375 and 0.3984375
33.
AREA PROBLEM Example 1
We could obtain better
estimates by increasing
the number of strips.
34.
AREA PROBLEM Example 1
The table shows the
results of similar
calculations (with a
computer) using n
rectangles, whose
heights are found with
left endpoints (Ln)
or right endpoints
35.
AREA PROBLEM Example 1
In particular, we see
that by using:
50 strips, the area lies
between 0.3234 and 0.3434
1000 strips, we narrow it
down even more—A lies
between 0.3328335 and
0.3338335
36.
AREA PROBLEM Example 1
A good estimate is
obtained by averaging
these numbers:
A≈
37.
AREA PROBLEM
From the values in the
table, it looks as if Rn
is approaching 1/3 as n
We confirm this in
the next example.
38.
AREA PROBLEM Example 2
For the region S in Example 1, show that
the sum of the areas of the upper
approximating rectangles approaches 1/3,
that is,
lim Rn 13
n
39.
AREA PROBLEM Example 2
Rn is the sum of the
areas of the n rectangles.
Each rectangle has width
1/n and the heights are
the values of the function
f(x) = x2 at the points
1/n, 2/n, 3/n, …, n/n.
That is, the heights are
(1/n)2, (2/n)2, (3/n)2, …,
(n/n)2.
40.
AREA PROBLEM Example 2
2 2 2 2
11 1 2 1 3 1 n
Rn ...
n n n n n n n n
1 1 2 2 2 2
2 (1 2 3 ... n )
n n
1 2 2 2 2
3 (1 2 3 ... n )
n
41.
AREA PROBLEM E. g. 2—Formula 1
Here, we need the formula for the sum of
the squares of the first n positive integers:
2 2 2 n(n 1)(2n 1)
2
1 2 3 ... n
6
Perhaps you have seen this formula before.
It is proved in Example 5 in Appendix E.
42.
AREA PROBLEM Example 2
Putting Formula 1 into our expression
for Rn, we get:
1 n(n 1)(2n 1)
Rn 3
n 6
(n 1)(2n 1)
2
6n
43.
AREA PROBLEM Example 2
(n 1)(2n 1)
So, we have: lim Rn lim
n n 6n 2
1 n 1 2n 1
lim
n 6
n n
1 1 1
lim 1 2
n 6
n n
1
12
6
1
3
44.
AREA PROBLEM
It can be shown that the lower
approximating sums also approach 1/3,
that is,
lim Ln 1
3
n
45.
AREA PROBLEM
From this figure, it
appears that, as n
increases, Rn becomes a
better and better
approximation to the
area of S.
46.
AREA PROBLEM
From this figure too, it
appears that, as n
increases, Ln becomes a
better and better
approximations to the
area of S.
47.
AREA PROBLEM
Thus, we define the area A to be the limit of
the sums of the areas of the approximating
rectangles, that is,
A lim Rn lim Ln 13
n n
48.
AREA PROBLEM
Let’s apply the idea of
Examples 1 and 2
to the more general
region S of the earlier
49.
AREA PROBLEM
We start by
subdividing S into n
S1, S2, …., Sn of equal
50.
AREA PROBLEM
The width of the interval [a, b] is b – a.
So, the width of each of the n strips is:
b a
x
n
51.
AREA PROBLEM
These strips divide the interval [a, b] into n
[x0, x1], [x1, x2], [x2, x3], . . . , [xn-1, xn]
where x0 = a and xn = b.
52.
AREA PROBLEM
The right endpoints of the subintervals are:
x1 = a + ∆x,
x2 = a + 2 ∆x,
x3 = a + 3 ∆x,
.
.
.
53.
AREA PROBLEM
Let’s approximate the i th
strip Si by
a rectangle with width ∆x
and height f(xi), which is
the value of f at the right
Then, the area of the i th
rectangle is f(xi)∆x.
54.
AREA PROBLEM
What we think of
intuitively as the area of
is approximated by the
sum of the areas of these
rectangles: Rn = f(x1) ∆x
+ f(x2) ∆x + … + f(xn) ∆x
55.
AREA PROBLEM
Here, we show this
approximation for
n = 2, 4, 8, and 12.
56.
AREA PROBLEM
Notice that this
approximation appears to
become better and better
as the number of strips
that is, as n → ∞.
57.
AREA PROBLEM
Therefore, we define
the area A of the region S
as follows.
58.
AREA PROBLEM Definition 2
The area A of the region S that lies
under the graph of the continuous function f
is the limit of the sum of the areas of
approximating rectangles:
A lim Rn
n
lim[ f ( x1 )x f ( x2 )x ... f ( xn )x]
n
59.
AREA PROBLEM
It can be proved that the limit in
Definition 2 always exists—since
we are assuming that f is continuous.
60.
AREA PROBLEM Equation 3
It can also be shown that we get the same
value if we use left endpoints:
A lim Ln
n
lim[ f ( x0 )x f ( x1 )x ... f ( xn 1 )x]
n
61.
SAMPLE POINTS
In fact, instead of using left endpoints or right
endpoints, we could take the height of the i th
rectangle to be the value of f at any number xi*
in the i th subinterval [xi - 1, xi].
We call the numbers xi*, x2*, . . ., xn*
the sample points.
62.
AREA PROBLEM
The figure shows
approximating rectangles
when the sample points
are not chosen to be
63.
AREA PROBLEM Equation 4
Thus, a more general expression for
the area of S is:
A lim[ f ( x1*) x f ( x2 *)x ... f ( xn *) x]
n
64.
SIGMA NOTATION
We often use sigma notation to write sums
with many terms more compactly.
For instance,
n
f ( x )x f ( x )x f ( x )x ... f ( x )x
i 1
i 1 2 n
65.
AREA PROBLEM
Hence, the expressions for area
in Equations 2, 3, and 4 can be written
as follows: n
A lim f ( xi )x
n
i 1
n
A lim f ( xi 1 )x
n
i 1
n
A lim f ( xi *) x
n
i 1
66.
AREA PROBLEM
We can also rewrite Formula 1 in
the following way:
n
2 n(n 1)(2n 1)
i
i 1 6
67.
AREA PROBLEM Example 3
Let A be the area of the region that lies under
the graph of f(x) = e-x between x = 0 and x = 2.
a. Using right endpoints, find an expression for A
as a limit. Do not evaluate the limit.
b. Estimate the area by taking the sample points
to be midpoints and using four subintervals
and then ten subintervals.
68.
AREA PROBLEM Example 3 a
Since a = 0 and b = 2, the width of
a subinterval is:
2 0 2
x
n n
So, x1 = 2/n, x2 = 4/n, x3 = 6/n, xi = 2i/n,
xn = 2n/n.
69.
AREA PROBLEM Example 3 a
The sum of the areas of the approximating
rectangles is:
Rn f ( x1 )x f ( x2 )x ... f ( xn )x
x1 x2 xn
e x e x ... e x
2/ n 2 4/ n 2 2n / n 2
e e ... e
n n n
70.
AREA PROBLEM Example 3 a
According to Definition 2, the area is:
A lim Rn
n
2 2/ n 4/ n 6/ n 2n / n
lim (e e e ... e )
n n
Using sigma notation, we could write:
2 n 2i / n
A lim e
n n
i 1
71.
AREA PROBLEM Example 3 a
It is difficult to evaluate this limit directly
by hand.
However, with the aid of a computer algebra
system (CAS), it isn’t hard.
In Section 5.3, we will be able to find A
more easily using a different method.
72.
AREA PROBLEM Example 3 b
With n = 4, the subintervals of equal width
∆x = 0.5 are:
[0, 0.5], [0.5, 1], [1, 1.5], [1.5, 2]
The midpoints of these subintervals are:
x1* = 0.25, x2* = 0.75, x3* = 1.25, x4* = 1.75
73.
AREA PROBLEM Example 3 b
The sum of the areas
of the four rectangles
4
M 4 f ( xi *)x
i 1
f (0.25)x f (0.75)x f (1.25)x f (1.75)x
0.25 0.75 1.25 1.75
e (0.5) e (0.5) e (0.5) e (0.5)
0.25 0.75 1.25 1.75
(e
1
2 e e e ) 0.8557
74.
AREA PROBLEM Example 3 b
With n = 10, the subintervals are:
[0, 0.2], [0.2, 0.4], . . . , [1.8, 2]
The midpoints are:
x1* = 0.1, x2* = 0.3, x3* = 0.5, …, x10* = 1.9
75.
AREA PROBLEM Example 3 b
A M 10
f (0.1)x f (0.3)x f (0.5)x ... f (1.9) x
0.2(e 0.1 e 0.3 e 0.5 ... e 1.9 )
0.8632
76.
AREA PROBLEM Example 3 b
From the figure, it
appears that
this estimate is better
than the estimate with n
= 4.
77.
DISTANCE PROBLEM
Now, let’s consider the distance problem:
Find the distance traveled by an object during
a certain time period if the velocity of the
object is known at all times.
In a sense, this is the inverse problem of the velocity
problem that we discussed in Section 2.1
78.
CONSTANT VELOCITY
If the velocity remains constant, then
the distance problem is easy to solve
by means of the formula
distance = velocity x time
79.
VARYING VELOCITY
However, if the velocity varies,
it’s not so easy to find the distance
We investigate the problem in the following
example.
80.
DISTANCE PROBLEM Example 4
Suppose the odometer on our car is
broken and we want to estimate the
distance driven over a 30-second time
81.
DISTANCE PROBLEM Example 4
We take speedometer
every five seconds
and record them
in this table.
82.
DISTANCE PROBLEM Example 4
In order to have the time and the velocity
in consistent units, let’s convert the velocity
readings to feet per second
(1 mi/h = 5280/3600 ft/s)
83.
DISTANCE PROBLEM Example 4
During the first five seconds, the velocity
doesn’t change very much.
So, we can estimate the distance traveled during that
time by assuming that the velocity is constant.
84.
DISTANCE PROBLEM Example 4
If we take the velocity during that time interval
to be the initial velocity (25 ft/s), then we
obtain the approximate distance traveled
during the first five seconds:
25 ft/s x 5 s = 125 ft
85.
DISTANCE PROBLEM Example 4
Similarly, during the second time interval,
the velocity is approximately constant, and
we take it to be the velocity when t = 5 s.
So, our estimate for the distance traveled
from t = 5 s to t = 10 s is:
31 ft/s x 5 s = 155 ft
86.
DISTANCE PROBLEM Example 4
If we add similar estimates for the other time
intervals, we obtain an estimate for the total
distance traveled:
(25 x 5) + (31 x 5) + (35 x 5)
+ (43 x 5) + (47 x 5) + (46 x 5)
= 1135 ft
87.
DISTANCE PROBLEM Example 4
We could just as well have used the velocity
at the end of each time period instead of
the velocity at the beginning as our assumed
constant velocity.
Then, our estimate becomes:
(31 x 5) + (35 x 5) + (43 x 5)
+ (47 x 5) + (46 x 5) + (41 x 5)
= 1215 ft
88.
DISTANCE PROBLEM Example 4
If we had wanted a more accurate
estimate, we could have taken velocity
readings every two seconds, or even
every second.
89.
DISTANCE PROBLEM
Perhaps the calculations in Example 4
remind you of the sums we used earlier
to estimate areas.
90.
DISTANCE PROBLEM
The similarity is
explained when we
a graph of the velocity
function of the car
and draw rectangles
whose heights are
the initial velocities for
each time interval.
91.
DISTANCE PROBLEM
The area of the first
rectangle is 25 x 5 = 125,
which is also our
estimate for the distance
traveled in the first five
In fact, the area of each
rectangle can be
interpreted as a distance,
because the height
represents velocity and
the width represents time.
92.
DISTANCE PROBLEM
The sum of the areas of
the rectangles is L6 =
1135, which is our initial
estimate for
the total distance
93.
DISTANCE PROBLEM
In general, suppose an object moves
with velocity
v = f(t)
where a ≤ t ≤ b and f(t) ≥ 0.
So, the object always moves in the positive
direction.
94.
DISTANCE PROBLEM
We take velocity readings at times
t0(= a), t1, t2, …., tn(= b)
so that the velocity is approximately constant
on each subinterval.
If these times are equally spaced, then
the time between consecutive readings is:
∆t = (b – a)/n
95.
DISTANCE PROBLEM
During the first time interval, the velocity
is approximately f(t0).
Hence, the distance traveled is
approximately f(t0)∆t.
96.
DISTANCE PROBLEM
Similarly, the distance traveled during
the second time interval is about f(t1)∆t
and the total distance traveled during
the time interval [a, b] is approximately
f (t0 )t f (t1 )t ... f (tn 1 ) t
n
f (ti 1 )t
i 1
97.
DISTANCE PROBLEM
If we use the velocity at right endpoints
instead of left endpoints, our estimate for
the total distance becomes:
f (t1 ) t f (t2 )t ... f (tn )t
n
f (ti )t
i 1
98.
DISTANCE PROBLEM
The more frequently we measure
the velocity, the more accurate our
estimates become.
99.
DISTANCE PROBLEM Equation 5
So, it seems plausible that the exact distance
d traveled is the limit of such expressions:
n n
d lim f (ti 1 )t lim f (ti ) t
n n
i 1 i 1
We will see in Section 5.4 that this is indeed true.
100.
Equation 5 has the same form as our
expressions for area in Equations 2 and 3.
So, it follows that the distance traveled
is equal to the area under the graph of
the velocity function.
101.
In Chapters 6 and 8, we will see that other
quantities of interest in the natural and social
sciences can also be interpreted as the area
under a curve.
Examples include:
Work done by a variable force
Cardiac output of the heart
102.
So, when we compute areas in this
chapter, bear in mind that they can
be interpreted in a variety of practical
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