Riemann Sum: ∫_a^b f(x) dx ≈ Σ f(x_k*) Δx ,   Δx = (b-a)/n

Choose method: left, right, midpoint, or trapezoid.

Function f(x)

Enter a function of x. Use ^ for exponent, * for multiplication, sin(), cos(), exp(), log(), sqrt().

x²

sin(x)

e⁻ˣ

4 - x²

√(4-x²)

1/(1+x²)

Lower limit (a)

Upper limit (b)

Number of subintervals (n)

1 to 200

Riemann sum method Left endpoint Right endpoint Midpoint Trapezoidal rule

Decimal places 4 digits 6 digits 8 digits 10 digits

Compute Riemann Sum Reset

Understanding Riemann Sums

Riemann sums are a family of methods to approximate the definite integral ∫_a^b f(x) dx. The interval [a, b] is divided into n subintervals of equal width Δx = (b−a)/n. On each subinterval [x_k−1, x_k], we choose a sample point x_k* and form a rectangle (or trapezoid) with height f(x_k*). The sum of the areas approximates the integral.

Common Rules (Sample Point Choices)

Method	Sample point xk*	Formula (Δx = (b−a)/n)	Geometric interpretation
Left Riemann sum	Left endpoint: a + (k−1)Δx	Δx·[f(a) + f(a+Δx) + … + f(b−Δx)]	Height of each rectangle is the left‑side value; underestimates increasing functions, overestimates decreasing functions.
Right Riemann sum	Right endpoint: a + kΔx	Δx·[f(a+Δx) + f(a+2Δx) + … + f(b)]	Height is right‑side value; overestimates increasing functions, underestimates decreasing functions.
Midpoint rule	Midpoint: a + (k−½)Δx	Δx·[f(a+½Δx) + f(a+1½Δx) + … + f(b−½Δx)]	Height at subinterval midpoints; often more accurate (error O(Δx²)).
Trapezoidal rule	—	(Δx/2)·[f(a) + 2f(a+Δx) + 2f(a+2Δx) + … + f(b)]	Uses trapezoids instead of rectangles; averages left and right sums, also O(Δx²) error.

Error Analysis & Convergence

For a continuous function, as n → ∞ (Δx → 0), all Riemann sums converge to the true integral. The speed of convergence depends on the method:

• Left/Right sums: error ≈ O(Δx) (first‑order). Doubling n roughly halves the error.
• Midpoint & Trapezoidal rules: error ≈ O(Δx²) (second‑order). Doubling n reduces error by a factor of about 4.

More precisely, if f is twice continuously differentiable, the midpoint rule error is about −(b−a)Δx² f''(ξ)/24, and trapezoidal error is about −(b−a)Δx² f''(ξ)/12 for some ξ in (a,b). Hence midpoint is often slightly better.

Why Use Riemann Sums?

• Numerical integration when an antiderivative is difficult or impossible (e.g., f(x) = e^−x²).
• Foundation of the definite integral – the limit of Riemann sums defines the integral.
• Applications in physics (work, distance, center of mass), probability (expected values), economics (consumer surplus), and engineering (average values).

Frequently Asked Questions

Quick Tips

• Use * for multiplication, e.g., 2*x, not 2x
• For power, use ^: x^3, sin(x)^2
• sqrt(x) = square root, exp(x) = e^x
• Larger n gives better approximation
• Use pi for π, e for Euler's number in limits