Riemann sums: rectangles, signed area and ∫_a^b f(x) dx

Year 12 Mathematics Advanced · S_n = h [f(x₁*) + … + f(x_n*)] with h = (b − a)/n · a rectangle below the axis counts negative · as n grows, S_n → F(b) − F(a)

Curve

Graph

curve rectangle above axis (+) below axis (−)

n strips 4

4 strips, width h = 0.75

Sample each strip at its

Signed (net) — the definite integral

S_n (rectangles) –

exact F(b) − F(a) –

closed form –

error S_n − exact –

Unsigned (total) — every rectangle counted positive

Σ h·|f(x*)| –

exact ∫ |f| dx –

The n sample heights

Each rectangle contributes h × f(x*). A negative height is a rectangle hanging below the axis: it subtracts from the signed sum and adds to the total.

Net or total?

Signed view: a rectangle below the axis is drawn downward and counts negative, exactly as ∫ f dx does. Switch to see every rectangle flipped upward, which is what "total distance" or "total area" means.

Three rules, one limit

Left and right sums bracket the integral on a monotone curve; the midpoint sum is usually closest. Try all three at the same n, then push n up: all three converge to F(b) − F(a).