Integral function and fundamental theorem

What is it about?

The integral function $I_a(x) = \int_a^x f(t)\,dt$ measures the signed area from $a$ to $x$ (green above, red below the $x$-axis); always $I_a(a) = 0$.

Fundamental theorem (for continuous $f$): $I_a'(x) = f(x)$ – the mean height $\frac{I_a(x+h) - I_a(x)}{h}$ of a strip lies between $\min f$ and $\max f$ on the strip and tends to $f(x)$ as $h \to 0$. For every antiderivative $F$: $\int_a^b f(x)\,dx = F(b) - F(a)$; a constant $C$ cancels.

Consequences: zeros of $f$ with a sign change are extrema of $I_a$, extrema of $f$ are points of inflection of $I_a$, and another lower limit only shifts the graph vertically. Integrating a rate of change gives the total change: speed → distance, power → energy, marginal cost → cost …

Try it

1. Parabola: press “Let the area grow”.
2. Strip h → 0: press “Let h tend to 0” and watch the secant.
3. Working: change the constant $C$ – the result stays the same.
4. Cubic: find maxima, minima and inflection points of $I_{-3}$; then move $a$.
5. Applications: car, solar panels, medicine, thunderstorm.

Tasks

Task 1: Evaluate

Evaluate $\int_{-1}^{2}(0.5x^2 - 1)\,dx$. Load task

Task 2: How far does the car travel?

Speed $3t$ for 5 s, then 15 s at $15\,\tfrac{\text{m}}{\text{s}}$, then 5 s of even braking. Load task

Task 3: Riemann sums

For $f(x) = x^2$ on $[0, 2]$ with $n = 4$: left, right, midpoint and trapezoid sums? Load task

Ideas for the classroom

• The views follow a possible lesson sequence: integral function → derivative (idea of proof) → evaluation → curve sketching of $I_a$ → Riemann sums. The applications connect integrals with units (kW · h = kWh).
• Common stumbling blocks: confusing the integral with the area when $f$ changes sign; equating integral functions with all antiderivatives; swapping $F(b) - F(a)$; forgetting that the theorem needs continuity (at a jump $I$ has a kink).