6.2 Approximating Areas with Riemann Sums

What is a Riemann Sum?

A Riemann sum approximates the area under a curve by summing areas of rectangles. Divide the interval [a,b] into n subintervals of width Δx, pick a sample point in each, and multiply the function value there by Δx. Types include left, right, and midpoint sums, depending on where the sample point is chosen in each subinterval.

Left, Right, and Midpoint Sums

Left sums use the left endpoint of each subinterval, often underestimating area for increasing functions. Right sums use the right endpoint, often overestimating for increasing functions. Midpoint sums use the middle of each subinterval and are generally more accurate. For decreasing functions, over/underestimate behavior reverses.

Trapezoidal Approximation

The trapezoidal rule averages left and right sums, using trapezoids instead of rectangles for a better fit to curved graphs. Formula: T = (Δx/2)[f(x₀) + 2f(x₁) + ... + 2f(xₙ₋₁) + f(xₙ)]. It's especially useful when data is given as a table rather than a formula.

Key terms

Riemann Sum

A method for approximating the area under a curve by summing the areas of rectangles.

Left Riemann Sum

Uses the function value at the left endpoint of each subinterval as rectangle height.

Right Riemann Sum

Uses the function value at the right endpoint of each subinterval as rectangle height.

Midpoint Sum

Uses the function value at the midpoint of each subinterval; generally more accurate than left/right sums.

Trapezoidal Rule

Approximates area using trapezoids; equals the average of the left and right Riemann sums: T = (Δx/2)[f(x₀)+2f(x₁)+...+f(xₙ)].

Try an AP-style question

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

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