AP Calculus AB — calculusStandard2 min read

Riemann Sums and Approximation

Approximate integrals using left, right, midpoint, and trapezoidal Riemann sums for AP Calculus AB.

Tutor AI Team2026-02-25T02:13:54.044Z

Riemann sums approximate definite integrals by dividing the area into rectangles or trapezoids.

Types

Left Riemann Sum (LRAM)

Use the left endpoint of each subinterval for height.

Right Riemann Sum (RRAM)

Use the right endpoint.

Midpoint (MRAM)

Use the midpoint of each subinterval.

Trapezoidal

$T_n = \frac{\Delta x}{2}[f(x_0) + 2f(x_1) + 2f(x_2) + ... + 2f(x_{n-1}) + f(x_n)]$

Over- and Under-estimates

Function	Left	Right	Midpoint	Trap
Increasing	Under	Over	—	Over
Decreasing	Over	Under	—	Under
Concave up	—	—	Under	Over
Concave down	—	—	Over	Under

Worked Example

$\int_0^4 x^2\,dx$ with 4 subintervals ( $\Delta x = 1$ ).

LRAM: $0 + 1 + 4 + 9 = 14$ . RRAM: $1 + 4 + 9 + 16 = 30$ . Trapezoidal: $\frac{1}{2}(0 + 2(1) + 2(4) + 2(9) + 16) = \frac{44}{2} = 22$ . Exact: $\frac{64}{3} \approx 21.33$ .

Practice Problems

1. Estimate $\int_1^5 \sqrt{x}\,dx$ using LRAM with 4 subintervals.
1. Is RRAM an over- or under-estimate for a decreasing function?

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Key Takeaways

✓
More subintervals → better approximation.
✓
Know which methods over/under-estimate.
✓
Trapezoidal often gives the best estimate.

Tags:

#ap#calculus#integration#riemann-sums#approximation

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