In the rectangle method, the function to integrate \( f \) is replaced by a piecewise constant function \( g(x) \) on each elementary interval \([x_{i-1}, x_i]\):

• Left rectangles: \( g(x) = f(x_{i-1}) \) for \( x \in [x_{i-1}, x_i] \)
• Right rectangles: \( g(x) = f(x_i) \) for \( x \in [x_{i-1}, x_i] \)
• Midpoint rectangles: \( g(x) = f\left(\frac{x_{i-1}+x_i}{2}\right) \) for \( x \in [x_{i-1}, x_i] \)

The associated integration formulas are:

Left rectangles: ∫_a^b f(x) dx ≈ Σ_i=1ⁿ (x_i - x_{i-1}) f(x_{i-1})
Right rectangles: ∫_a^b f(x) dx ≈ Σ_i=1ⁿ (x_i - x_{i-1}) f(x_i)
Midpoint rectangles: ∫_a^b f(x) dx ≈ Σ_i=1ⁿ (x_i - x_{i-1}) f((x_{i-1}+x_i)/2)

If the subdivision is uniform with step \( h = \frac{b-a}{n} \), then:

Left rectangles: ∫_a^b f(x) dx ≈ h Σ_i=0^n-1 f(x_i)
Right rectangles: ∫_a^b f(x) dx ≈ h Σ_i=1ⁿ f(x_i)
Midpoint rectangles: ∫_a^b f(x) dx ≈ h Σ_i=1ⁿ f((x_{i-1}+x_i)/2)

Error

The left and right rectangle methods are order 0. If the first derivative of \( f \) is bounded by \( M \):

Left rectangles: | ∫_a^b f(x) dx - h Σ_i=0^n-1 f(x_i) | ≤ (b-a)/2 × h × sup |f'(x)|
Right rectangles: | ∫_a^b f(x) dx - h Σ_i=1ⁿ f(x_i) | ≤ (b-a)/2 × h × sup |f'(x)|

The midpoint rectangle method is order 1. If \( f \in C^2([a,b]) \):

| ∫_a^b f(x) dx - h Σ_i=1ⁿ f((x_{i-1}+x_i)/2) | ≤ (b-a)/24 × h² × sup |f''(x)|

Numerical Example

Evaluate numerically ∫₀² sin(x) dx with n=3 (h=2/3):

Midpoints:

Results:

• Left rectangles: R_g = (2/3)(0 + 0.6184 + 0.9705) ≈ 1.0586
• Right rectangles: R_d = (2/3)(0.6184 + 0.9705 + 0.9093) ≈ 1.6646
• Midpoint rectangles: R_m = (2/3)(0.3270 + 0.8415 + 0.9950) ≈ 1.5410

The exact integral is I(f) = 1 − cos(2) ≈ 1.4161. Relative errors are:

• Error(R_g) = |1.4161 − 1.0586| ≈ 0.3575 → 25.23%
• Error(R_d) = |1.6646 − 1.4161| ≈ 0.2485 → 17.55%
• Error(R_m) = |1.5410 − 1.4161| ≈ 0.1249 → 8.82%