Riemann Sums & the Definite Integral

You’ll be able to

Approximate area under a curve with left, right, and midpoint Riemann sums
Interpret the definite integral as a limit of Riemann sums
Predict whether a sum overestimates or underestimates based on monotonicity

Approximating area with rectangles

To estimate the area under a curve, slice the interval into subintervals and build a rectangle on each. A left Riemann sum uses the left endpoint of each subinterval for the height, a right sum uses the right endpoint, and a midpoint sum uses the middle. Each rectangle’s area is height × width, and summing them approximates the region — the more rectangles, the better the estimate.

Over- and underestimates

On an increasing function, a left sum uses the lower left heights and underestimates, while a right sum uses the higher right heights and overestimates. On a decreasing function the roles flip. The trapezoidal sum averages left and right and is usually more accurate. Knowing the direction of the error lets you bound the true value without more computation.

The definite integral as a limit

As the number of rectangles n → ∞ and their widths shrink to zero, the Riemann sum converges to the exact area — the definite integral ∫[a to b] f(x) dx. It represents the net signed area between the curve and the x-axis: area above the axis counts positive, area below counts negative. The integral is the precise value the approximating sums are reaching toward.

Riemann sum and the integral

∫[a to b] f(x) dx = lim(n→∞) Σ f(xᵢ*)·Δx, where Δx = (b − a)/n

A left sum takes xᵢ* at left endpoints, a right sum at right endpoints. The limit is the exact net signed area.

Worked example

Estimate the area under f(x) = x² on [0, 2] using a right Riemann sum with 4 rectangles of equal width.

1.Width of each rectangle: Δx = (2 − 0)/4 = 0.5. Right endpoints are x = 0.5, 1.0, 1.5, 2.0.
2.Heights f(x) = x²: f(0.5) = 0.25, f(1) = 1, f(1.5) = 2.25, f(2) = 4.
3.Multiply each height by Δx = 0.5 and add: 0.5(0.25 + 1 + 2.25 + 4) = 0.5(7.5).
4.That gives 3.75.

Answer: The right sum estimate is 3.75. Since x² is increasing, this overestimates the true area (∫[0 to 2] x² dx = 8/3 ≈ 2.667).

Tip

On an increasing function, remember "left is low, right is high": the left sum underestimates and the right sum overestimates. Reverse it for a decreasing function. This lets you state over/underestimate with confidence.

Checkpoint

For an increasing function on [a, b], a left Riemann sum will:

Checkpoint

What is Δx for a Riemann sum of ∫[1 to 7] f(x) dx using 3 equal subintervals?

On the exam

AP problems often give data in a table and ask for a specific Riemann sum. Read carefully whether left, right, or midpoint endpoints are required, and note that subintervals may have unequal widths — compute each rectangle’s width separately then.

Answer the 2 checkpoints as you read.