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CALCULUS 1 PROBLEMS &
SOLUTIONS
6.2.1
The Definite Integral
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Review
1. Upper And Lower Riemann Sums
Let f be a continuous function on the closed interval
[a,
b],
n
a positive integer, Pn the
regular partition
of order n of [a, b],
and Dx =
(b
– a)/n.
We use an example of n = 10 in Figs. 1, 2, and 3.
Let Rn(
f,
a,
b)
be a Riemann sum for f on [a, b]
relative to Pn and
using arbitrary points of the subintervals, ie:
where ci is an arbitrary point
of the ith subinterval [xi–1, xi],
for i = 1, 2, ..., n. See Fig.
1. Since the
points of the subintervals used are arbitrary, Rn(
f,
a,
b)
is called the general Riemann sum for f on
[a,
b].
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Fig. 1 |
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Fig. 2 |
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Fig. 3
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Because f is continuous on each subinterval, it has a
maximum and a minimum there (see Section
1.2.2
Theorem 1). Let ui and
li
be points of [xi–1, xi]
where f(ui)
is the maximum and f(li)
the minimum of f
on [xi–1, xi].
Let Un( f,
a,
b)
and Ln( f,
a,
b)
be the Riemann sums for f on [a,
b]
relative to Pn, Un
using the points ui's
and Ln using the points li's,
ie:
as shown in Fig. 2 for Un
and in Fig. 3 for Ln.
For each i = 1, 2, ..., n, we have f(li)
£
f(ci)
£
f(ui).
Thus, multiplying each side by Dx
> 0 and summing it up for i from 1 to n
we obtain:
Ln( f,
a,
b)
£
Rn(
f,
a,
b)
£
Un(
f,
a,
b).
That's the reason for the following definition.
Definition 1
¾¾¾¾¾¾¾¾¾¾¾¾
Un(
f,
a,
b)
is called the upper Riemann sum for f on
[a,
b],
and Ln( f,
a,
b)
is called the lower
Riemann sum for f on [a,
b].
¾¾¾¾¾¾¾¾¾¾¾¾
Remarks 1
· Recall that each rectangle above the x-axis
contributes its area to, thus increases, the Riemann
sum, while each rectangle below the x-axis
contributes the negative of its area to, thus decreases,
the Riemann sum.
· For a rectangle above the x-axis,
if its height increases (decreases), then the Riemann sum
increases (decreases).
· For
a rectangle below the x-axis, if its height increases
(decreases), then the Riemann sum
decreases (increases).
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2. The Definite Integral
In Section
6.1.3, every function f is positive on an interval
of the form [a, b], and limn®¥ Rn(
f,
a,
b)
is
the area A of the plane region bounded by the graph of f,
the x-axis, the vertical line x
= a,
and the
vertical line x = b. Here, f
can be positive or negative on [a, b]
or some subset of it, as illustrated in
Figs. 1, 2, and 3. In these figures, limn®¥ Rn(
f,
a,
b)
isn't A. Rather, as shown in Fig. 4, it's the area
under the positive-valued parts of the graph of f and above
the x-axis plus the negative of, thus minus,
the area above the negative-valued parts of the graph of f
and under the x-axis. In Fig. 4, it equals
area A1 – area A2. That quantity is defined in
the following definition.
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Fig. 4
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Definition 2
¾¾¾¾¾¾¾¾¾¾¾¾
¾¾¾¾¾¾¾¾¾¾¾¾
Terminology
· ò is
called the integral sign. It's an elongated letter S, because it
represents the limit of a sum.
· a is called the lower limit
of integration and b the upper limit of integration,
both referred to
collectively as the limits of
integration.
· f is called the integrand.
· x is called the variable of
integration.
· Recall that dx is the differential
of x
(see Section
2.6 Definitions 1).
Remarks
2
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Fig. 5 |
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3. Definite Integrals And
Areas
i. If f ³ 0 on [a,
b],
then A is defined to be I:
A
=
I.
ii. If f < 0 on
[a,
b],
then A = – I (A
> 0 because I < 0).
iii. For general f
like the one in Fig. 4, I equals the total area of the
parts of R lying above the x-axis
minus the total area of the parts
of R
lying under the x-axis.
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Problems & Solutions
Solution
The points xi's
are x0 = 0, x1 = 0 + b/n
= b/n,
x2 = 0 + 2b/n
= 2b/n,
..., xn = 0 + nb/n
= nb/n.
In
each subinterval [xi–1, xi]
the maximum of f occurs at xi
and the minimum at xi–1. Thus:
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Solution
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Solution
a.
b.
c.
d.
