Areas between curves, RMS values and probability

Lesson 9 introduced the definite integral as an accumulated total and showed how the Fundamental Theorem of Calculus turns its evaluation into antidifferentiation. With that tool in hand, many quantities that would otherwise need laborious summation become single calculations. This lesson develops several applications that appear throughout engineering: reading and using accumulation functions, finding the area between two curves, exploiting symmetry, computing root-mean-square (RMS) values, which determine heating in electrical equipment, and using integrals to describe probabilities, which underlie reliability, quality control and risk. The integrals here require only the antiderivatives you already know; the more advanced techniques of integration, and applications such as volumes and centroids, follow in MATH-101.

Accumulation functions

An accumulation function adds up a rate from a fixed starting point to a variable end:

By the Fundamental Theorem, \(A'(x) = f(x)\): the rate at which the total grows is the value of the integrand at the moving end. This lets you read the behaviour of \(A\) from a graph of \(f\), even when no formula is available. Where \(f > 0\), \(A\) increases; where \(f < 0\), \(A\) decreases; \(A\) has a local maximum where \(f\) changes from positive to negative; and \(A\) is concave up where \(f\) is increasing. All of the curve-sketching ideas of Lesson 6 apply, with \(f\) playing the role of the derivative.

The area between two curves

If \(f(x) \ge g(x)\) on \([a, b]\), the area of the region between their graphs is the integral of the difference, top minus bottom:

Think of slicing the region into thin vertical strips of width \(dx\) and height \(f(x) - g(x)\), and adding them up. If the curves cross, find the intersection points, split the interval there, and integrate the top minus the bottom on each piece. When a region is more naturally bounded by curves written as \(x = h(y)\), slice horizontally instead and integrate right minus left with respect to \(y\).

Using symmetry

Symmetry can halve the work or remove it entirely. If \(f\) is even, \(f(-x) = f(x)\), its graph is symmetric about the \(y\)-axis, and \(\int_{-a}^{a} f(x)\,dx = 2\int_0^a f(x)\,dx\). If \(f\) is odd, \(f(-x) = -f(x)\), the areas on the two sides cancel, and \(\int_{-a}^{a} f(x)\,dx = 0\). For example, \(\int_{-\pi}^{\pi} x^3 \cos x\,dx = 0\) immediately, because \(x^3\cos x\) is odd.

Periodic functions have a similar property: the integral over any complete period is the same, wherever the period starts. In particular, the integral of \(\sin\) or \(\cos\) over a whole number of periods is zero. This fact, combined with the identities for products of sines and cosines, is the foundation of Fourier analysis, which decomposes signals into frequencies (ELE courses).

Root-mean-square values

The average value of a sinusoidal voltage or current over a cycle is zero, yet a resistor connected to an AC supply certainly heats up. The heating depends on the square of the current, since power in a resistor is \(P = i^2 R\). The useful measure is therefore the root-mean-square value: the square root of the average of the square.

A steady direct current equal to \(I_{\text{RMS}}\) produces the same average heating as the varying current. For a sinusoid \(i = I_p \sin\omega t\), the identity \(\sin^2\theta = \tfrac{1}{2}(1 - \cos 2\theta)\) gives an average of \(\sin^2\) over a period of exactly one half, so:

Household voltages are quoted as RMS values: the 120 V supply in North America has a peak of about 170 V, and a 230 V supply a peak of about 325 V.

Integrals and probability

Many engineering quantities are uncertain: the lifetime of a component, the strength of a batch of concrete, the time between data packets, the error in a measurement. Such a quantity is modelled as a continuous random variable \(X\) with a probability density function \(f(x)\). The density itself is not a probability; instead, the probability that \(X\) falls between \(a\) and \(b\) is the area under the density:

A density must be non-negative and have total area 1, \(\int_{-\infty}^{\infty} f(x)\,dx = 1\), since \(X\) must take some value. The mean, or expected value, is the balance point of the distribution, \(\mu = \int x f(x)\,dx\), a calculation that often needs the integration techniques of MATH-101. STAT-235 develops these ideas fully.

The exponential distribution and reliability

ELE-470, Lesson 2, described components that fail at a constant rate \(\lambda\). Their lifetime \(T\) has the exponential density \(f(t) = \lambda e^{-\lambda t}\) for \(t \ge 0\). The probability of failure by time \(t\) is:

and the probability of surviving beyond \(t\), the reliability, is \(R(t) = e^{-\lambda t}\). The mean lifetime turns out to be \(1/\lambda\), the mean time between failures.

The uniform and normal distributions

A uniform distribution on \([a, b]\) has constant density \(1/(b - a)\): every value in the interval is equally likely, as with the rounding error of a digital measurement. The normal (Gaussian) distribution, with its bell-shaped density \(f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-(x - \mu)^2/(2\sigma^2)}\), describes many measured quantities, but its integral has no elementary antiderivative (Lesson 8), so its probabilities are computed numerically and tabulated. About 68 percent of its area lies within one standard deviation \(\sigma\) of the mean, 95 percent within two, and 99.7 percent within three, figures used throughout quality control.