Approximating Areas with Riemann Sums

Area under a curve is hard because the top edge is curved, not straight. Rectangles fix that: chop the interval into n equal strips of width Delta x, build a rectangle on each strip, and add up the areas. The height of each rectangle has to come from somewhere, so you pick a rule. A left sum uses the function value at the left edge of each strip, a right sum uses the right edge, and a midpoint sum uses the point in the middle. Each choice draws a flat top on a curved region, so each one misses, just in different places. As Delta x shrinks and n grows, the gaps between rectangle top and curve shrink too, and all three methods converge to the same exact area - the one the definite integral gives you.

Whether a method overshoots or undershoots depends on where the function is increasing or decreasing, not on some fixed rule. A left sum underestimates an increasing function because every left edge sits below the curve on its strip, but on a decreasing function that same left sum overestimates, since now the left edge sits above the curve. So 'left sum = underestimate' is not a law, just something that happens to be true when f is increasing. Concavity tells a different, more reliable story: trapezoids overestimate area when the curve is concave up and underestimate when it is concave down, because a straight trapezoid edge cuts across a cup shape on the outside. Midpoint sums do the opposite - they underestimate when concave up and overestimate when concave down, since the midpoint rectangle tucks neatly under the curve's bulge.

This topic is the bridge into the rest of the unit. Once you see that finer rectangles squeeze the approximation toward one exact number, that limiting number is defined as the definite integral, which the next topics learn to compute exactly instead of estimate.