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Math Analysis 1

The definite integral: fundamental theorem, areas and applications in circuit theory

Riemann sums, properties, mean value theorem, fundamental theorem of calculus, areas between curves, charge, energy, mean and RMS values, numerical integration.

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The definite integral starts from a geometric question, how large is the area under a graph, but it answers many physical ones: the charge carried by a current, the energy absorbed over a time interval, the mean and RMS value of a signal. The fundamental theorem of calculus connects it to antiderivatives and makes it computable.

Definition with Riemann sums

Split [a, b] into n small intervals of width \Delta x = \frac{b-a}{n}, choose a point \xi_k in each and add up the areas of the rectangles:

    \[ S_n = \sum_{k=1}^{n} f(\xi_k)\,\Delta x. \]

If, as the subdivision gets finer, S_n tends to the same limit whatever points are chosen, f is integrable and the limit is \int_a^b f(x)\,dx. Continuous functions are integrable, and so are functions with finitely many jumps. Areas below the x axis count with a minus sign.

Properties

  • Linearity: \int_a^b (\alpha f + \beta g) = \alpha\int_a^b f + \beta\int_a^b g.
  • Additivity: \int_a^b f = \int_a^c f + \int_c^b f.
  • Monotonicity: if f \le g then \int_a^b f \le \int_a^b g; in particular \left|\int_a^b f\right| \le \int_a^b |f|.
  • Symmetry: if f is odd, \int_{-a}^{a} f = 0; if it is even, \int_{-a}^{a} f = 2\int_0^a f.

Mean value theorem for integrals

If f is continuous on [a, b] there is c such that \int_a^b f(x)\,dx = f(c)\,(b - a). The quantity \frac{1}{b-a}\int_a^b f is the mean value of f: in electronics it is the DC component of a signal.

The fundamental theorem of calculus

Let f be continuous. The integral function F(x) = \int_a^x f(t)\,dt is differentiable and F'(x) = f(x): the integral is an antiderivative. Consequently, if G is any antiderivative,

    \[ \int_a^b f(x)\,dx = G(b) - G(a) = \Big[G(x)\Big]_a^b. \]

Example: \int_0^\pi \sin x\,dx = [-\cos x]_0^\pi = 1 + 1 = 2. Without this theorem you would need to compute a limit of sums every time.

With a substitution the limits change too: \int_0^1 2x\,e^{x^2}dx, with t = x^2, becomes \int_0^1 e^t\,dt = e - 1. By parts: \int_a^b f'g = [fg]_a^b - \int_a^b fg'.

Areas

The area between two graphs with g \le f on [a, b] is \int_a^b (f - g)\,dx. For the area between y = x and y = x^2: the curves meet at 0 and 1, and on that interval x \ge x^2, so the area is \int_0^1 (x - x^2)\,dx = \frac12 - \frac13 = \frac16. If a function changes sign and you want the geometric area, integrate |f|, that is split the interval at the points where f vanishes.

Applications in circuit theory

  • Charge: q = \int_{t_1}^{t_2} i(t)\,dt. The voltage across a capacitor is v(t) = v(0) + \frac1C\int_0^t i(\tau)\,d\tau.
  • Energy: W = \int_{t_1}^{t_2} v(t)\,i(t)\,dt. The energy stored in a capacitor charged to V is \int_0^V C v\,dv = \frac12 CV^2.
  • RMS value: V_{\text{rms}} = \sqrt{\frac1T\int_0^T v^2(t)\,dt}. For v = V_m\sin\omega t it gives \frac{V_m}{\sqrt2}: the 230 V of the European mains are an RMS value, with a peak of about 325 V.
  • Mean value of a full-wave rectified sine: \frac{1}{\pi}\int_0^\pi V_m\sin x\,dx = \frac{2V_m}{\pi} \approx 0.637\,V_m.

A longer physical application, the charge and discharge of a capacitor studied with integration, is in the in-depth article linked to this syllabus topic.

Numerical integration (outline)

When the antiderivative is not elementary, go back to the definition: the trapezoidal rule replaces rectangles with trapezoids, Simpson’s rule with parabolic arcs. With Simpson and only 10 intervals, \int_0^1 e^{-x^2}dx \approx 0.74682, already correct to five decimal places.

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