Riemann Sum — Left Endpoint Approximation

A Riemann sum approximates the area under a curve by stacking rectangles. The "left-endpoint" version uses the function value at the left side of each subinterval to determine the rectangle's height. For an increasing function, this systematically underestimates the true area.

The Setup

Divide $[a, b]$ into $n$ equal subintervals of width:

$\Delta x = \dfrac{b - a}{n}$

The left endpoints are $x_i = a + i\,\Delta x$ for $i = 0, 1, \ldots, n - 1$. The left Riemann sum is:

$L_n = \sum_{i=0}^{n-1} f(x_i)\,\Delta x$

Step-by-Step Solution

Given: $f(x) = x^{2}$, $a = 0$, $b = 4$, $n = 8$.

Find: The left-endpoint Riemann sum $L_8$.

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Step 1 — Compute $\Delta x$.

$\Delta x = \dfrac{b - a}{n} = \dfrac{4 - 0}{8} = 0.5$

Step 2 — List the left endpoints.

$x_i = 0 + i(0.5) \;\;\text{for}\;\; i = 0, 1, \ldots, 7$

$x_0, x_1, \ldots, x_7 = 0,\;0.5,\;1.0,\;1.5,\;2.0,\;2.5,\;3.0,\;3.5$

Step 3 — Evaluate $f(x_i) = x_i^{2}$ at each.

$f(0) = 0, \;\; f(0.5) = 0.25, \;\; f(1) = 1, \;\; f(1.5) = 2.25,$

$f(2) = 4, \;\; f(2.5) = 6.25, \;\; f(3) = 9, \;\; f(3.5) = 12.25$

Step 4 — Sum the function values.

$\sum_{i=0}^{7} f(x_i) = 0 + 0.25 + 1 + 2.25 + 4 + 6.25 + 9 + 12.25 = 35.0$

Step 5 — Multiply by $\Delta x$.

$L_8 = 0.5 \times 35.0 = 17.5$

Step 6 — Compare to the exact value.

$\int_{0}^{4} x^{2}\,dx = \dfrac{x^{3}}{3}\bigg|_{0}^{4} = \dfrac{64}{3} \approx 21.333$

The error is $21.333 - 17.5 = 3.833$, or about 18% underestimation. This makes sense: $f(x) = x^{2}$ is increasing, so the left endpoint of each subinterval is the lowest value of $f$ on that subinterval — every rectangle sits below the curve.

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Answer: The left-endpoint Riemann sum with $n = 8$ rectangles is

$\boxed{\;L_8 = 17.5\;}$

This underestimates the exact area $\dfrac{64}{3} \approx 21.333$ by approximately 3.83 square units (≈ 18% error). To improve the estimate, use more rectangles, switch to right or midpoint sums, or use Simpson's rule.

Try It

• Adjust n (number of rectangles) — watch the error shrink as $n$ grows.
• Toggle show exact area to see the true area highlighted in green.
• The HUD shows the computed $L_n$ alongside the exact value $64/3$ for comparison.
• Notice how every rectangle's top-left corner touches the curve — that's the defining feature of a left Riemann sum.