Riemann sum

Parametric equations

On the interval $[0, 1]$ , take $n$ rectangles of equal width. The height is the left endpoint $f (x_{i})$ :

Area \approx i = 0 \sum n - 1 f (x_{i}) Δ x, Δ x = \frac{1}{n}, x_{i} = i Δ x

The function in this figure:

f (x) = 1 + 0.65 sin (k π x + t) cos (π x)

n sets how many rectangles there are. k changes the shape of the integrand. t moves the function in time.

Number of partitions n: add rectangles and watch the area sum approach the region under the curve
Wave frequency k: change the rise and fall of the integrand, and watch the error at the same n
Time speed ω: keep the wave and the rectangle heights updating

The height of each rectangle is the value of the function on that subinterval, and the sum of the areas is a Riemann sum.
As $n$ grows, the sum usually gets closer to the definite integral, for an integrable function.
The sharper the bend, the more visible the error at a finite $n$ .