Riemann Sum

Approximating the area under a curve by summing the areas of thin rectangles — the geometric intuition behind the definite integral.

Riemann sums approximate area with rectangles

Rule

Rectangles: 4

Estimate

R₄ = 3.750

Exact area is 2.667.

Definition

A Riemann Sum approximates the area under a curve by dividing the region into rectangles and summing their areas.

For a function $f$ on $[a, b]$ , divide into $n$ subintervals of width $\Delta x = \frac{b - a}{n}$ . Choose a sample point $x_i^*$ in each subinterval $[x_{i-1}, x_i]$ . The Riemann sum is:

$R_n = \sum_{i=1}^{n} f(x_i^*) \cdot \Delta x$

Three common choices for $x_i^*$ :

Left Riemann Sum: $x_i^* = x_{i-1}$ (left endpoint of each subinterval)
Right Riemann Sum: $x_i^* = x_i$ (right endpoint)
Midpoint Rule: $x_i^* = \frac{x_{i-1} + x_i}{2}$

Right Riemann sum for x²

Approximate $\int_0^2 x^2\, dx$ using a right Riemann sum with $n = 4$ .

$\Delta x = 0.5$ . Right endpoints: $x_1 = 0.5$ , $x_2 = 1.0$ , $x_3 = 1.5$ , $x_4 = 2.0$ .

$R_4 = 0.5[(0.5)^2 + (1)^2 + (1.5)^2 + (2)^2] = 0.5[0.25 + 1 + 2.25 + 4] = 0.5(7.5) = 3.75$

The exact value is $\frac{8}{3} \approx 2.667$ . Our approximation overestimates because $x^2$ is increasing.

Try it

Approximate $\int_0^1 x\, dx$ using a left Riemann sum with $n = 4$ . Compare to the exact answer.

Solution

$\Delta x = 0.25$ . Left endpoints: $0, 0.25, 0.5, 0.75$ .

$L_4 = 0.25[0 + 0.25 + 0.5 + 0.75] = 0.25 \cdot 1.5 = 0.375$

Exact: $\int_0^1 x\, dx = \frac{1}{2} = 0.5$ . Error $= 0.125$ . The left sum underestimates because $f(x) = x$ is increasing.

Related concepts

Needs first

Limits

Fundamental Theorem of Calculus Integrals

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