6.2 Approximating Areas with Riemann Sums

Rectangles

A Riemann sum approximates the area under a curve using rectangles: left, right, or midpoint sums (or trapezoids). Using more rectangles gives a more accurate estimate.

Over/under

For an increasing function, a left sum underestimates and a right sum overestimates (the reverse for a decreasing function).

Key terms

Riemann sum

Approximates area under a curve with rectangles.

Types

Left, right, midpoint sums (and trapezoids).

More rectangles

Give a more accurate estimate.

Left sum (increasing f)

Underestimates the area.

Right sum (increasing f)

Overestimates the area.

Try an AP-style question

Using 2 equal subintervals, what is the left Riemann sum for f(x) = x² on [0, 2]?

1. 5
2. 8/3
3. 2
4. 1