Math · Calculus I · Worked example

Left, right and midpoint sums for x²

Estimate ∫₀¹ x² dx with n = 2 rectangles, using left, right and midpoint heights. Compare each estimate with the exact value 1/3.

\int_{0}^{1} x^{2} d x, n = 2

Choose the partition

With n = 2, Δx = (1 − 0)/2 = 1/2, so the endpoints are 0, 1/2 and 1.

Δ x = \frac{1 - 0}{2} = \frac{1}{2}

Left sum

The left endpoints are 0 and 1/2, with heights 0 and 1/4.

L_{2} = \frac{1}{2} (0 + \frac{1}{4}) = \frac{1}{8}

Right sum

The right endpoints are 1/2 and 1, with heights 1/4 and 1.

R_{2} = \frac{1}{2} (\frac{1}{4} + 1) = \frac{5}{8}

Midpoint sum

The midpoint sum M₂ samples at 1/4 and 3/4: square each midpoint, add the heights and multiply by the width 1/2.

\frac{1}{2} (\frac{1}{16} + \frac{9}{16}) = \frac{5}{16}

Compare with the exact value

An antiderivative of x² is x³/3, so the Fundamental Theorem gives the exact value 1/3 ≈ 0.333. Because x² increases on [0, 1], the left sum 0.125 is too low and the right sum 0.625 too high. The midpoint sum 0.3125 misses by only 1/48 ≈ 0.021.

\int_{0}^{1} x^{2} d x = \frac{1^{3}}{3} - \frac{0^{3}}{3} = \frac{1}{3}

Result

L₂ = 1/8 = 0.125, R₂ = 5/8 = 0.625 and M₂ = 5/16 = 0.3125, against the exact 1/3 ≈ 0.333.

Your turn

Estimate ∫₀² x dx with two midpoint rectangles. Why is the estimate exact?

Show the answer and explanation

M₂ = 2, which equals the integral.

Δx = 1 and the midpoints are 0.5 and 1.5, so M₂ = 1·0.5 + 1·1.5 = 2. For a straight line, each rectangle gains exactly the small triangle it loses on the other side of its midpoint, so the midpoint rule is exact.

\begin{array}{l} 1 \cdot 0.5 + 1 \cdot 1.5 = 2 \\ \int_{0}^{2} x d x = \frac{2^{2}}{2} - \frac{0^{2}}{2} = 2 \end{array}

Keep exploring

In Riemann Sums, raise n from 2 to 4 and then 8. The midpoint sum becomes 0.328125 and then 0.33203125: each doubling of n makes the error four times smaller.

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