Riemann Sum — Right Endpoint Approximation

The right-endpoint Riemann sum uses the function value at the right side of each subinterval. For an increasing function like $f(x) = x^{2}$, this systematically overestimates the true area — every rectangle sticks above the curve.

The Setup

Same partition as the left sum, but we evaluate $f$ at the right endpoint $x_i = a + i\,\Delta x$ for $i = 1, 2, \ldots, n$:

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

Step-by-Step Solution

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

Find: The right-endpoint Riemann sum $R_8$.

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Step 1 — Compute $\Delta x$.

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

Step 2 — List the right endpoints.

$x_i = 0.5,\; 1.0,\; 1.5,\; 2.0,\; 2.5,\; 3.0,\; 3.5,\; 4.0$

Step 3 — Evaluate $f(x_i) = x_i^{2}$.

$f(0.5) = 0.25, \;\; f(1) = 1, \;\; f(1.5) = 2.25, \;\; f(2) = 4,$

$f(2.5) = 6.25, \;\; f(3) = 9, \;\; f(3.5) = 12.25, \;\; f(4) = 16$

Step 4 — Sum the function values.

$\sum_{i=1}^{8} f(x_i) = 0.25 + 1 + 2.25 + 4 + 6.25 + 9 + 12.25 + 16 = 51.0$

Step 5 — Multiply by $\Delta x$.

$R_8 = 0.5 \times 51.0 = 25.5$

Step 6 — Compare to the exact value.

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

The error is $25.5 - 21.333 = 4.167$, or about 20% overestimation. The right sum is slightly worse than the left sum (which underestimated by 18%), because $f$ grows faster as $x$ grows — the error in the rightmost rectangle alone is huge.

Step 7 — Average the left and right sums (Trapezoidal Rule).

A neat observation: averaging $L_n$ and $R_n$ gives a much better estimate. This average is the Trapezoidal Rule:

$T_n = \dfrac{L_n + R_n}{2} = \dfrac{17.5 + 25.5}{2} = 21.5$

That's only 0.167 away from the exact value $64/3 \approx 21.333$ — less than 1% error with the same number of subdivisions!

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

$\boxed{\;R_8 = 25.5\;}$

This overestimates the exact area $\dfrac{64}{3} \approx 21.333$ by 4.17 square units (≈ 20% error). The trapezoidal average $(L_n + R_n)/2 = 21.5$ is almost exactly right.

Try It

• Adjust n — watch how both the left and right errors shrink toward the exact value $64/3$.
• Toggle show exact area to see the true region in faint green.
• Notice how every rectangle's top-right corner touches the curve — that's the defining feature of a right Riemann sum.