Here is a brief list of the key elementary properties of the definite
integral for continuous functions. The proofs of these properties follow
directly from the definition of the definite integral based on the
Euler (left-hand endpoint) sums. Each property can be interpreted either
in a motion / velocity context or, when the functions involved have
only nonnegative values, in an area context. You will find these
interpretations provided with each statement to help you make sense of
these properties. They should help to convince you of the validity of
the properties as well as suggest how to develop more rigorous arguments
as proofs. The proofs based on the definition are outlined in the
problems as exercises in understanding and using the definition of the
definite integral. 0. a) [The Reverse order / alternating property.] $\int_a^b P(x) dx = - \int _b^a P(x) dx$.
This property is sometimes used as a notational definition when $a > b $.
Motion Interpretation: The net change in position of an object moving
with velocity $P$ will have the same magnitude but opposite depending on
whether we consider the change as occurring from time $a$ to time $b$
or from time $b$ to time $a$.
b) [The integral over a point is zero.] $\int_a^a P(x) dx = 0$.
Motion Interpretation: The net change in position of an object moving when there is no change in time is 0.
Geometry Interpretation: When $P(x)>0$, the region described by the
region between $X=a$ and $X=a$ is in fact a line segment which has area
$0$. (The width for this region is $0$!)
Motion Interpretation: If the velocity scale is changed by a factor of α
the net change in position will be changed by precisely the same
factor.
Geometry Interpretation: If the heights of the upper boundary curve of a
region are rescaled by a factor of α, the area of the resulting region
with this new bounding curve will be related to the area of the original
region by precisely the same multiplicative factor.
2. [ The Sum Property.] $\int_a^b P(x) \pm Q(x) dx = \int_a^b P(x) dx \pm \int_a^b Q(x) dx$ .
[1 and 2 are called the Linearity Properties of the definite integral.]
3.[The Monotonicity Properties.]
a) If $P(x) ≥ 0$ for all $x ε [a,b]$ then $\int_a^b P(x) dx ≥ 0$.
Motion Interpretation: If the velocity of an object is always
non-negative then the net change in the object's position will also be
non-negative.
Geometric Interpretation: If the function $P$ is non-negative, the area
of the region enclosed by the graph of $P$, the lines $X=a, X=b$, and
the $X$-axis is non-negative.
b) If $R(x)≥Q(x)$ for all $x ε
[a,b]$ then $\int_a^b R(x) dx ≥ \int_a^b Q(x) dx$.
Motion Interpretation: If the velocity of one object, $R(x)$, is always
greater than or equal to the velocity of a second object, $Q(x)$, then
the first object will have a net change in position that is greater than
or equal to the net change in position of the second object.
Geometry Interpretation: Assuming $R(x)≥Q(x)≥0$, the region bounded by
the graph of $R$ will contain the region bounded by the graph of $Q$, so
the corresponding area for the function $R$ will be greater than the
area for the function $Q$.
4. [The Integral of a Constant.] $\int_a^b K dx = K (b-a)$.
Motion Interpretation: The net change in position for an object moving
at a constant velocity will be the product of the velocity with the
change in time.
Geometry Interpretation: Area of the related region when $P(x)= K$ is
the area of a reactangle with base length $b-a$ and height $K$, that is,
$K (b-a)$.
For the sake of completeness of our list, we list one property here that is more subtle. It
makes sense in both the geometric and motion interpretations. It's proof
is not difficult for continuous functions once we have proven the
Fundamental Theorems of Calculus.
5. [The Additive Property.] $\int_a^b P(x) dx = \int_a^c P(x) dx + \int_c^b P(x) dx$ for any $a, b,$ and $c$.
Geometry Interpretation: If $P(x) > 0$ and $a < c < b$ , then
the area of the regions above the $X$-axis and below the graph of
$Y=P(X)$ between the lines $X=a$ and $X=b$, $\int_a^b P(x)
dx$ , is the sum of the areas of the regions enclosed by the $X$-axis,
the graph of $Y=P(X), X=a$ and $X=c$, $\int_a^c P(x) dx$ , and by the
$X$-axis, the graph of $Y=P(X), X=c$ and $X=b$, $\int_c^b P(x)
dx$ .
See Figure V.B.1.
Motion Interpretation: Assume $a < c < b$. The net change in
position of an object moving with velocity $P$ between time $a$ and
time $b$, $\int_a^b P(x) dx$, is the sum of the net changes in position
between time $a$ and time $c$, $\int_a^c P(x) dx$, and between time $c$
and time $b$, $\int_c^b P(x) dx$. See Figure V.B.1.
Figure V.B.1.
These properties allow some determination of definite integrals based on
knowledge of the components that make up the integrand. Although the
calculus is not as easy as that of the derivative here are some examples
of how the properties just discussed can be used for an elementary
calculus for the definite integral. Example VI.B.1. Suppose $P$ and $Q$ are continuous functions on $[a,b]$
and $\int_a^b P(x) dx= 3$ while $\int_a^b Q(x) dx =-2$ . Find the
following definite integrals:
Solutions: For a) and b) we use the scalar property so
a) $\int_a^b 5 P(x) dx = 5 \int_a^b P(x) dx = 5 \cdot 3 = 15$ ;
b) $\int_a^b -5 P(x) dx = -5 \int_a^b P(x) dx = (-5) (-2) = 10.$
For c) and d) we use linearity more fully, so
c)$\int_a^b P(x) - Q(x) dx = \int_a^b P(x)dx - \int_a^b Q(x)dx = 3 - (-2) = 5$ ;
d) $\int_a^b 5P(x)+2Q(x)dx = 5\int_a^b P(x)dx + 2\int_a^b Q(x)dx = 5(3)+2(-2) = 11$.
Finally for e) note that $\int_a^b Q(t)dt = \int_a^b Q(x) dx = -2$ ,
2. $\int_1^5 2P(x) - Q(x) dx$; $\int_1^5 3P(x) - 2Q(x) dx$ .
3. $\int_1^5 P(x) + 2 dx$ ; $\int_1^5 P(x) - x dx$ ; $\int_1^5 2 - P(x) dx$ .
4. $\int_1^5 2P(x) - x dx$; $\int_1^5 2Q(x) - 1 dx$ .
5. Find $k$ so that $\int_1^5 P(x) - k dx = 0; \int_1^5 Q(x) - k dx = 0$.
6. Find $k$ so that $\int_1 k P(x) dx = 1; \int_1 k Q(x) dx = 1$.
7. Use the area interpretation of the definite integral together with
the fundamental theorem of calculus of chapter IV.D to verify property 5
when $P(x) = x^2 , a = 1, b=5$ and $c=4$. Draw a graph that
illustrates the area interpretation of property 5 for this example.
8. Use the Euler sums in the definition of the definite integral to justify property 1 .
9. Use the Euler sums in the definition of the definite integral to justify property 2 .
10. Use the Euler sums in the definition of the definite integral to justify property 3 .
11. Use the Euler sums in the definition of the definite integral to justify property 4 .
12. Assume property 3a. Prove property 3b by considering the function $P(x)= R(x) - Q(x)$ and using linearity.
13. a) Show that $\frac 12 < \int_1^2 \frac 1x ~dx < 1$. [Hint: Use monotonicity .]
b) Show that $\frac 13 < \int_1^{\frac 32} \frac 1x ~dx < \frac 12$.
c) Show that $\frac 14 < \int_{\frac 32}^2 \frac 1x ~dx < \frac 13$.
d) Using parts b and c show that $\frac 7{12} < \int_1^2 \frac 1x~ dx < \frac 56$.
14. Show that $\frac 12 < \int_0^1 \frac 1{1+x^2} dx < 1$.
15.Give a geometric and a velocity interpretation for property 4.
16. Probability density functions. Suppose f is a probability density
function for a continuous random variable X and the probability that X
is between 2 and 4 is 1/3 while the probability that X is between 3 and 5
is 2/5 and the probability the X is between 2 and 5 is 1/2.
a) Express the information given about the probabilities as information
about the values of the related definite integrals of the density
function. Draw a figure to visualize the information using the area
interpretation of the definite integral. Draw a figure to visualize the
information using the motion interpretation of the definite integral.
b) Use the Additive property of integrals to find the probability that X
is between 5 and 7. Explain the relation of your conclusion to the
figures you drew in part a).
c) Find the probability that X is between 4 and 5.
d) Based on the information you are given and can infer about the random
variable $X$, discuss in which of the following intervals the value
of $X$ is most likely to fall: [2,3], [3,4], or [4,5].
e) Suppose the range of the random variable $X$ is [1, 6] and $\int_1^2~f(x)~dx~=~ \frac 18$.
i) What is the probability that $X$ is between 5 and 6?
ii) In what interval will the median value of $X$ be located?
17. Average probability value minimizes an integral. Suppose f is a
probability density function for a continuous random variable $X$ with
range [a,b]. Let $V(k)= \int_a^b ~(f(x) - k)^2~dx$ where $k$ is any real
number. $V$ is called the squared variation of the (density) function
$f$ from the number $k$.
a) Use the properties of the definite integral to explain why the
function $V$ is a quadratic polynomial function of the variable $k$,
i.e., there are constants $A,B, C$ where $V(k)~=~Ak^2 + Bk +C$.
[Hint: The numbers $A,B,$ and $C$ will be expressed as definite
integrals.]
b) Analyze the function $V$ using the derivative to show that the value
of $V$ is smallest at the number $k_*$ where $k_* = \frac{ \int_a^b
~f(x)~dx} {b-a}$ or $ k_* \cdot {(b-a)} = {\int_a^b ~f(x)~dx}$ .
18. Average value minimizes an integral. Suppose $f$ is a
continuous function defined over the compact interval $[a,b]$. Let
$V(k)= \int_a^b ~(f(x) - k)^2~dx$ where $k$ is any real
number. $V$ is called the squared variation of the function $f$
from the number $k$.
a) Use the properties of the definite integral to explain why the
function $V$ is a quadratic polynomial function of the variable $k$,
i.e., there are constants $A, B, C$ where $V(k)~=~Ak^2 + Bk +C$.
[Hint: The numbers $A,B$, and $C$ will be expressed as definite
integrals.]
b) Analyze the function $V$ using the derivative to show that the value
of $V$ is smallest at the number $k_*$ where $k_* = \frac{\int_a^b
~f(x)~dx}{b-a}$ or $k_* \cdot {(b-a)} = {\int_a^b ~f(x)~dx}$ .
c) Consider $f$ as the velocity function for some object in motion.
Interpret the number $k_*$ found in part b) using the motion context to
explain the statement that the value of $V$ is smallest when $k$ is the
average velocity of the object.
d) Consider the graph of $f$ forming the upper boundary of a region in
the plane bounded also by the lines $X=a, X=b$, and the $X$-axis.
Interpret $k_*$ as the height of a rectangle to explain the statement
that the value of $V$ is smallest when the area of that region equals
the area of the rectangle with height $k_*$ and base $(b-a)$.
19. Probability density functions. Suppose $f$ is a probability
density function for a random variable $X$ and the probability that $X$
is between $A$ and $C$ is $p$ while the probability that $X$ is between
$B$ and $D$ is $q$ and the probability the $X$ is between $A$ and $D$ is
$r$, where $A<B<C<D$.
a) Express the information given about the probabilities as information
about the values of the related definite integrals of the density
function.
b) Find the probability that $X$ is between C and D.
c) Find the probability that $X$ is between B and C.
20.Project.? Mean of random variable minimizes variation. Introduce variation in earlier applications of definite integral.
Define Expected Value of $Y=f(X)$. Prove linearity of $E(Y)$. Expand variation to show variation $= E( X^2) - (E(X))^2$