Trapezoidal Sums

Rectangles force a flat top on every piece, so they either miss the curve's rise or overshoot it. A trapezoid tilts its top edge to connect the actual function values at both ends of the interval, so it hugs the curve far more closely. That tilt is the whole improvement: instead of picking one height (left, right, or midpoint) and pretending the function is constant, you use both endpoint heights and let the shape lean between them.

Mechanically, each trapezoid on a subinterval of width delta x has parallel sides equal to f at the left endpoint and f at the right endpoint. Area of a trapezoid is the average of the two parallel sides times the width, so each piece contributes (f(x_i-1) + f(x_i))/2 times delta x. Add up all the pieces and you get the trapezoidal sum. Because adjacent trapezoids share an endpoint height, the heights in the middle each get counted twice when you expand the sum, which is where the familiar pattern (f(x₀) + 2f(x₁) + 2f(x₂) + ... + f(x_n)) times delta x/2 comes from.

Whether the sum overshoots or undershoots the true integral depends entirely on concavity, not on which side you sample from. A chord connecting two points on a concave-up curve sits above the curve, so trapezoids overestimate. A chord on a concave-down curve sits below the curve, so trapezoids underestimate. If concavity switches inside the interval, the errors partly cancel, which is exactly why trapezoidal sums sit next to Riemann sums and later connect to error bounds and to Simpson's rule, which replaces straight chords with parabolas for an even tighter fit.