Integrals

Where this is used

Integration is adding things up — but adding up infinitely many infinitely small pieces. That single idea gives you the area under a curve, the volume of a tank, the total number of cases in an outbreak, and the total energy a device has consumed. It is also the exact reverse of differentiation, which is why it sits at the base of nearly all of science and engineering.

Start gently

Integration adds up the areas of very thin rectangles. The idea is exactly the one you used when you found an area by counting squares in primary school — the squares have just been made infinitely fine.

Key points

The indefinite integral

The indefinite integral runs differentiation backward: if F'(x) = f(x), then ∫f(x)dx = F(x) + C. The C is the constant of integration. It has to be there because a constant vanishes when you differentiate, so the original function could have had any constant attached to it.

The definite integral (area)

A definite integral gives the area a curve encloses between x = a and x = b — signed area, so anything below the x-axis counts as negative. To evaluate it, find an antiderivative, plug in the upper limit, and subtract its value at the lower limit.

The fundamental theorem of calculus

Differentiation and integration undo each other — the discovery Newton and Leibniz are famous for. Integrate speed and you get distance travelled; differentiate distance and you get speed back. That symmetry is what makes physics and engineering calculable at all: measure a rate and you can recover the total, or measure a total and you can recover the rate.

Area between two curves

To find the area trapped between two curves f(x) and g(x), subtract the lower curve from the upper one and integrate the difference across the interval where they overlap.

Integration in the real world

Integrate a power reading in kW over 24 hours and you get total energy in kWh. Integrate instantaneous speed over time and you get distance travelled. Integrate a probability density and you get a probability. That last one is why integration keeps showing up in AI and data science: every question about how likely a value is comes down to the area under a distribution.

See it drawn

Area as a count of squares (primary school)

Integration shrinks these squares without limit to fill the region under a curve.

Finding the area under y = x²

From 0 to 3 the area is ∫₀³ x² dx = 9 — the limit of adding up rectangular approximations.

Jobs that use this

Aerospace engineer$140k

Actuary$150k

Power systems engineer$116k

Start gently

Key points

See it drawn

Jobs that use this

Members-only from here

Comments