The integral as signed area (rectangle sums)

What is it about?

Split $[a; b]$ into $n$ strips of width $\Delta x = \frac{b-a}{n}$. The lower sum uses the smallest, the upper sum the largest function value in each strip; the area lies in between. As $n \to \infty$ both converge to the integral:

Areas below the x-axis count negatively (signed area).

Try it

1. Press Double n several times.
2. Drag the limits $a$ and $b$.
3. Sine from $0$ to $2\pi$. Signed area

Exercises

Exercise 1: By hand

Compute lower and upper sum of $x^2$ on $[0; 2]$ with $n = 4$. Load exercise

Ideas for the classroom

• Use: Grade 12/13 (definite integral).
• Common stumbling blocks: treating the integral and the area as the same even when the graph goes below the x-axis.