MATH 104, Sections 8.7 and 8.8 | Source: Rimmer lecture notes
Difficulty: Intermediate to Advanced | Prerequisites: All integration techniques from Sections 8.1 through 8.4, limits at infinity, L'Hopital's Rule
Sections 8.7 and 8.8 are where all the integration techniques you have learned come together and get applied to problems with infinite limits, singularities, or probabilistic interpretations. Improper integrals extend the definite integral to cases where the interval is infinite or the integrand blows up. The question is no longer "what is the value" but "does a finite value even exist?" Section 8.8 then uses improper integrals as the foundation of continuous probability, defining probability density functions, expected values, and standard deviation.
An improper integral replaces the infinite limit (or the singularity) with a variable, evaluates the resulting proper integral, and takes a limit. If the limit is finite, the integral converges; otherwise it diverges. A probability density function is a non-negative function whose total integral is 1, and its improper integrals give probabilities, expected values (mean), and variance.
Improper integral
A definite integral where at least one limit is infinite, or where the integrand has a discontinuity within the interval. In simple terms, you are trying to measure the area of a region that stretches to infinity or has a spike.
Convergent / Divergent
An improper integral converges if the limiting value exists and is finite. It diverges if the limit does not exist or is infinite.
Indeterminate form
An expression like 0/0 or infinity/infinity that arises when evaluating a limit. These forms do not have an automatic value and require further analysis (typically L'Hopital's Rule).
L'Hopital's Rule
A technique for evaluating limits of indeterminate forms 0/0 or infinity/infinity. Replace the limit of f(x)/g(x) with the limit of f'(x)/g'(x), provided the new limit exists.
Probability density function (PDF)
A function f(x) >= 0 whose total integral over all real numbers equals 1. The probability that a random variable falls between a and b is the integral of f(x) from a to b.
Mean (expected value)
The "average" outcome for a continuous random variable: mu = integral from negative infinity to infinity of x f(x) dx.
Standard deviation
A measure of spread: sigma = sqrt of the integral of (x minus mu)^2 f(x) dx over all x.
Exponential density function
The PDF f(x) = (1/mu) e^(-x/mu) for x >= 0, used to model waiting times and lifetimes. Its mean is mu and, remarkably, its standard deviation also equals mu.
Type 1: Infinite upper or lower limit
Infinite upper limit: Replace infinity with b, evaluate the integral from a to b, then take the limit as b approaches infinity.
\int_a^\infty f(x)\, dx = \lim_{b \to \infty} \int_a^b f(x)\, dxInfinite lower limit: Replace negative infinity with a, evaluate from a to b, then take the limit as a approaches negative infinity.
Both limits infinite (doubly improper): Split the integral at any convenient point c, then evaluate two separate improper integrals. Both must converge for the whole integral to converge.
Type 2: Discontinuity in the interval
If f(x) has a vertical asymptote at some point c in [a, b], split the integral at c and take one-sided limits:
\int_a^b f(x)\, dx = \lim_{t \to c^-} \int_a^t f(x)\, dx + \lim_{t \to c^+} \int_t^b f(x)\, dxKey limits to know:
If r > 0 is rational, then lim as x approaches infinity of 1/x^r = 0.
For rational functions: if degree of numerator < degree of denominator, the limit at infinity is 0. If degrees are equal, the limit is the ratio of leading coefficients.
Direct Comparison Theorem:
If 0 <= f(x) <= g(x) for all x >= a, then:
If integral of g(x) converges, so does integral of f(x).
If integral of f(x) diverges, so does integral of g(x).
In simple terms: a smaller function under a convergent one must converge, and a bigger function over a divergent one must diverge.
When evaluating limits that arise in improper integrals, you often encounter indeterminate forms.
The two indeterminate forms that L'Hopital's Rule handles directly:
0/0: both numerator and denominator approach zero.
infinity/infinity: both numerator and denominator approach infinity (or negative infinity).
The rule:
\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}provided the right-hand limit exists. You may apply the rule repeatedly if the result is still indeterminate.
Important: L'Hopital's Rule applies only to quotients in the 0/0 or infinity/infinity form. Other indeterminate forms (0 times infinity, 1^infinity, 0^0, infinity^0, infinity minus infinity) must first be rewritten as a quotient before the rule can be used.
Definition of a PDF:
A function f(x) is a probability density function if:
f(x) >= 0 for all x.
The integral of f(x) from negative infinity to infinity equals 1.
The probability that the random variable X falls between a and b is:
P(a \le X \le b) = \int_a^b f(x)\, dxMean (expected value):
\mu = \int_{-\infty}^{\infty} x\, f(x)\, dxMedian: The value m such that the integral of f(x) from negative infinity to m equals 1/2. Half the area under the curve lies on each side of the median.
Variance and standard deviation:
\sigma^2 = \int_{-\infty}^{\infty} (x - \mu)^2\, f(x)\, dx \qquad \sigma = \sqrt{\sigma^2}The exponential density function:
For modelling waiting times, lifetimes, and decay:
f(x) = \begin{cases} 0 & x < 0 \\ \frac{1}{\mu}\, e^{-x/\mu} & x \ge 0 \end{cases}The mean is mu and the standard deviation is also mu. This is a distinctive feature of the exponential distribution.
The normal density function (bell curve):
The PDF is f(x) = (1 / (sigma sqrt(2 pi))) e^(-(x minus mu)^2 / (2 sigma^2)). This integral has no closed-form antiderivative, so probabilities are computed from tables or software. You should know what it looks like and that it is symmetric about mu.
Topic | Formula | Notes |
|---|---|---|
Improper integral (infinite upper) | lim as b approaches infinity of integral from a to b of f(x) dx | Converges if the limit is finite |
Doubly improper | Split at c: two separate limits, both must converge | Choice of c does not matter |
L'Hopital's Rule | lim f(x)/g(x) = lim f'(x)/g'(x) | Only for 0/0 or infinity/infinity |
PDF requirement | integral of f(x) over all x = 1, and f(x) >= 0 | Defines a valid density |
Mean | mu = integral of x f(x) dx | Expected value |
Variance | sigma^2 = integral of (x minus mu)^2 f(x) dx | Spread around the mean |
Exponential PDF | f(x) = (1/mu) e^(-x/mu) for x >= 0 | Mean = mu, std dev = mu |
Comparison: convergent | 0 <= f <= g and integral of g converges implies integral of f converges | Smaller under convergent |
Comparison: divergent | 0 <= f <= g and integral of f diverges implies integral of g diverges | Bigger over divergent |
Improper integrals are how physicists compute the total energy radiated by a star over its lifetime (infinite time horizon) and how engineers determine whether a signal has finite total energy. The exponential density function models the time between arrivals in a queue (how long you wait for the next bus, the next customer, the next radioactive decay event). The normal distribution underpins virtually all of statistics.
Students try to evaluate an improper integral by plugging in infinity directly, writing things like "1/infinity = 0." You must use a limit with a finite variable approaching infinity.
When both limits are infinite, students sometimes take a single limit from negative infinity to infinity. You must split the integral at some finite point and evaluate two separate limits.
Students apply L'Hopital's Rule to fractions that are not in an indeterminate form. If the denominator is not approaching 0 or infinity alongside the numerator, the rule does not apply.
In probability, students forget that f(x) is a density, not a probability. The value f(3) = 0.5 does not mean there is a 50% chance of getting 3. You must integrate over an interval to get a probability.
Expect at least one convergence/divergence problem using the comparison theorem. Know the benchmark: integral of 1/x^p from 1 to infinity converges if p > 1 and diverges if p <= 1.
L'Hopital's Rule is tested both on its own and as a step inside an improper integral evaluation.
Probability questions typically ask you to verify that a function is a valid PDF (integrate and get 1), compute a probability over an interval, find the mean, or find the median.
The exponential distribution is a favourite because its integrals are tractable and it combines improper integrals with probability in one problem.
True or False: You can evaluate an improper integral by directly substituting infinity into the antiderivative.
False. You must use a limit.
Fill in the blank: L'Hopital's Rule applies only when the limit is in the ______ or ______ indeterminate form.
0/0 or infinity/infinity.
True or False: For the exponential distribution, the standard deviation equals twice the mean.
False. The standard deviation equals the mean (sigma = mu).
Fill in the blank: The integral of 1/x^p from 1 to infinity converges when p ______.
is greater than 1.
Q: Determine whether the integral from 1 to infinity of 1/x^2 dx converges or diverges. If it converges, find its value.
A: Replace infinity with b: integral from 1 to b of x^(-2) dx = [-1/x] from 1 to b = -1/b + 1. As b approaches infinity, -1/b approaches 0, so the integral converges to 1.
Q: Evaluate lim as x approaches infinity of x^2 / e^x.
A: This is infinity/infinity. Apply L'Hopital's Rule: lim 2x / e^x (still infinity/infinity). Apply again: lim 2 / e^x = 0.
Q: The PDF f(x) = (1/39) x^2 for 2 <= x <= 5, and 0 otherwise. Find the median.
A: Solve integral from 2 to m of (1/39) x^2 dx = 1/2. That gives (1/39)[x^3/3] from 2 to m = 1/2, so (m^3 minus 8) / 117 = 1/2, giving m^3 = 66.5, so m = (66.5)^(1/3), approximately 4.05.
Q: Use the comparison theorem to determine whether the integral from 1 to infinity of (2 + e^(-x)) / x^2 dx converges.
A: For x >= 1, 0 < (2 + e^(-x)) / x^2 <= 3/x^2. The integral of 3/x^2 from 1 to infinity converges (p = 2 > 1). By comparison, the given integral converges.
Improper integrals build directly on every integration technique from Chapter 8 (the antiderivative step is the same; the new element is the limit). The probability material in Section 8.8 connects forward to statistics courses and to moment-generating functions. The exponential distribution reappears in differential equations (it is the solution to dy/dx = -ky) and in the study of Markov chains and queueing theory.
Improper integral, convergent integral, divergent integral, infinite limits of integration, vertical asymptote, comparison test, direct comparison theorem, L'Hopital's Rule, indeterminate form, 0/0, infinity/infinity, probability density function, PDF, expected value, mean, median, variance, standard deviation, exponential distribution, normal distribution, bell curve, p-integral, p-test, MATH 104, Calculus II, Rimmer, Penn State Math 104