The Calculus Primer (2011)
Part XIV. The Definite Integral
Chapter 54. THE DEFINITE INTEGRAL AND ITS LIMITS
14—7. The Limits of Integration. It is easily shown that interchanging the limits of a definite integral is equivalent to changing the sign of the definite integral.
In other words, if the area “generated” in going from a to b is arbitrarily considered positive, then the area generated in going from b to a is negative, and vice versa.
Moreover, it is also readily seen that the definite integral is a function of its limits. Thus,
In other words, the value of the definite integral depends upon its limits as well as upon the nature of the function which comprises the integrand.
14—8. Breaking-up or Combining Intervals of Integration. From the figure, it will be seen that
and hence, by addition, we have:
then, by comparing the right-hand members of the last two equalities, it is clear that
The relationship is quite general. A definite integral can be broken up into any number of separate definite integrals in this manner; likewise, any number of separate definite integrals can be combined, provided the integrands are identical, by combining the limits. In fact, in the illustration, the limit b could just as well have been outside the interval from a to c; by giving attention to the signs of the areas as well as to their magnitudes, the relationship is still valid.
14—9. Improper Integrals. In the discussion thus far, the limits of integration have been finite quantities. If, however, either the upper or the lower limit, or both, should be infinite, the integral may or may not exist. For example, suppose that in the definite integral
the upper limit b → + ∞, that is, b increases indefinitely, while the lower limit a remains constant. The succession of values taken by the integral may then approach some definite, particular limiting value; if it does, this limiting value is taken as the “integral of ø(x) from x = a to x = + ∞,” and is designated by the expression
However, the succession of values assumed by the integral as b→ ∞ may not approach a limit; in this case, the expression given in equation (2) has no meaning, and we say that the integral does not exist.
We may then set forth the following relations by way of definition:
Here, in equations  and , the value of the expression on the left side is said to exist only if the limit on the right side exists.
Here, in equation , the integral on the left exists only if both integrals on the right exist.
EXAMPLE 1. Find .
EXAMPLE 2. Find
In other words, in this case, the
is infinite, or does not exist; hence the definite integral does not exist.
EXAMPLE 3. Find
This result being absurd, it is clear that the integral has no meaning and does not exist.
EXAMPLE 4. Find
= log ∞ − log 0 = ∞ − (−∞) = ∞.
Thus the integral is meaningless.
EXAMPLE 5. Find the total area under the curve
Since the X-axis is an asymptote, the limits of integration will of necessity be + ∞ and − ∞. As OS = x → ∞, arc tan as a limit; hence The entire area under the curve, since the Y-axis is an axis of symmetry, is therefore twice the area to the right of the Y-axis, or
Evaluate each of the following, if the integral exists:
14—10. Interpretation of an Integral. We have seen that a definite integral may be taken to represent an area. It does not follow, however, that every integral can only represent an area; it may represent the length of an arc along a curve. Or it may be possible to interpret an integral in terms of a physical quantity rather than a geometric magnitude. The meaning of an integral thus depends upon the nature of the quantities which are represented by the variables x and y. If for example, the variable x represents time and the variable y represents velocity, then the integral represents distance traveled. In fact, many physical quantities may be represented by integrals—for example, force, work, fluid pressure, center of gravity, moments of inertia, etc.
It is true, however, that even when an integral stands for a physical quantity, it may be represented geometrically as an area. In this connection, it may be interesting to note how the integral is useful in defining the mean value of a function. Thus, if y = f(x) is a given function, then the mean value of f(x) for the interval from x = a to x = b is given by
For, as the figure shows,
furthermore, if the area of rectangle AKTB, on AB, or (b − a), as a base, is equal in area to the area ARSB, then the
In other words, if M is so located that area RKM = area MTS, then MN is the mean value of f(x) for the interval from x = a to x = b.
14—11. Change of Limits when Changing the Variable. When evaluating a definite integral by the method of substitution, it is not necessary to go back to the original variable after the substitution has been made and the integration has been performed; it is only necessary to change the limits of integration to correspond to the change in the variable, as shown below.
EXAMPLE 1. Evaluate
Solution. Let then x = z2 — 1, and dx = 2z dz. Also, when x = 15, z = 4, and when x = 8, z = 3.
using formula , page 372, we have
EXAMPLE 2. Evaluate
Solution. Let x = a sin θ; then = a cos θ; also, dx = a cos θ dθ. Now when sin θ, or sin therefore θ = π/6. Again, when x = 0, a sin θ = 0, and θ = 0.
using formula , page 377, we have