Applications of the definite integral
Definite integrals give areas, volumes, work, accumulated charge and the energy a signal delivers to a load.
01Areas between curves
The area enclosed between two curves is the integral of the difference, the upper one minus the lower one, between the points where they intersect:
Between y = x and y = x²: they intersect at x = 0 and x = 1, and over that stretch the line is on top.
A = ∫₀¹ (x − x²) dx = [x²/2 − x³/3]₀¹ = 1/2 − 1/3 = 1/6.
02Volumes, lengths and surfaces
| Quantity | Formula | The idea |
|---|---|---|
| Volume by disks | V = π∫ab [f(x)]² dx | Rotating about the x-axis, each slice is a disk of radius f(x) |
| Volume by shells | V = 2π∫ab x·f(x) dx | Rotating about the y-axis, each slice is a thin shell of radius x |
| Arc length | L = ∫ab √(1 + [f′(x)]²) dx | Each tiny piece is the hypotenuse of dx and dy |
| Surface of revolution | S = 2π∫ab f(x)·√(1 + [f′]²) dx | Each tiny piece generates a ring |
The method is always the same, and it is worth remembering more than the formulas: cut into small pieces, write the contribution of a generic piece, and add up with an integral. This is also how you compute the work of a variable force, W = ∫F dx, the mass of a rod of variable density or the center of gravity of a part.
03Where ½CV² and ½LI² come from
The two energy formulas used from memory in electronics are, in fact, one-line definite integrals:
The same reasoning explains why charging a capacitor through a resistor dissipates exactly the same energy as the amount that ends up stored, ½CV², regardless of the resistor’s value: you integrate the power dissipated over the whole transient and that is the number you get. It is a result that surprises you the first time and that decides the efficiency of switching power supplies.
04Average and RMS values of any waveform
The definite integral gives the two numbers used to describe any periodic signal, and two more that are hardly ever taught and are used all the time: the form factor (RMS over rectified average) and the crest factor (peak over RMS), which tell how far the wave departs from a sine wave and how much headroom an amplifier needs.
The four numbers are computed by numerical integration over one full period, just as a true RMS instrument does.
05More applications in electronics
- Power supply ripple. The ripple voltage of a capacitor filter comes from integrating the load current during the time the diode does not conduct: ΔV = I·Δt/C. It is the same computation as in the regulated power supply.
- Average power. The power that actually heats a component is the average value of v·i over a period, not the product of the average values. With out-of-phase signals, cos φ appears.
- Battery charge. Capacity in ampere-hours is ∫i dt: the area under the current curve.
- Total harmonic distortion. It is defined with integrals of the square of each harmonic: it is a ratio between RMS values.
- Radiation dose, accumulated consumption, wear: any quantity that accumulates over time is a definite integral.
06In the lab
Compute the area between y = 4 − x² and the x-axis, and the volume of the solid it generates when rotated about the x-axis. Check the volume against the paraboloid formula if you know it.
Charge a capacitor of known value through a resistor and record v(t) and i(t). Numerically compute ∫v·i dt up to the end of the transient and compare with ½CV². Also compute the energy dissipated in the resistor and check that the two are equal.
Measure the average value, RMS value, form factor and crest factor of three signals from the generator with a digital oscilloscope, and compare with the values computed in the lab above. Discuss what an average-responding multimeter would read in each case.
07Common mistakes
- Subtracting the wrong way round when computing areas between curves and getting a negative area.
- Not looking for the crossing points and using arbitrary limits.
- Forgetting the π in volumes of revolution, or the square of the radius.
- Computing average power as the product of the average values of voltage and current.
- Using the factor 1.11 for any waveform: it is the form factor of the rectified sine wave.
- Integrating over an incomplete period and getting average values that mean nothing.
08Self-assessment
Area between y = x² and y = 2x.
They intersect at 0 and 2; the line is on top: ∫₀² (2x − x²) dx = 4 − 8/3 = 4/3.
Volume when y = √x, with 0 ≤ x ≤ 4, is rotated about the x-axis.
V = π∫₀⁴ x dx = π·8 = 8π ≈ 25.1.
How much energy does a 470 µF capacitor charged to 25 V store?
½·470·10−6·625 ≈ 0.147 J. Enough to give a noticeable spark when short-circuited: that is why they are discharged before being handled.
A square wave of 10 V peak with a 25% duty cycle between 0 and 10 V. Average and RMS value?
Average: 10·0.25 = 2.5 V. RMS: 10·√0.25 = 5 V. The RMS value is much larger than the average, and that is why a PWM signal heats more than its average value suggests.
What is the crest factor and what is it for?
The ratio of the peak to the RMS value. It indicates how much voltage headroom an amplifier or power supply needs to avoid clipping: for the sine wave it is √2, and for audio or data signals it can reach 4 or more.
09Further reading
- James Stewart. Single Variable Calculus: Early Transcendentals. 7th ed. (Spanish edition, “Cálculo de una variable. Trascendentes tempranas”), Cengage Learning, 2012. Chapter 6 goes through areas, volumes, work and average value with the “cut and add up” method.
- Charles K. Alexander and Matthew N. O. Sadiku. Fundamentals of Electric Circuits. 5th ed. (Spanish edition, “Fundamentos de circuitos eléctricos”), McGraw-Hill, 2013. Average and RMS values, stored energy and average power, with the integrals set up.
- Louis Leithold. The Calculus 7. 7th ed. (Spanish edition, “El cálculo”), Oxford University Press. Many geometric and physical application problems solved in detail.