Question

If f(x) = 3x2 − 8x, 0 ≤ x ≤ 3, evaluate the Riemann sum with n = 6, taking the sample points to be right endpoints.

Answer 1

Split up the interval [0, 3] into 6 subintervals,

[0, 1/2], [1/2, 1], [1, 3/2], [3/2, 2], [2, 5/2], [5/2, 3]

The right endpoints are given by the arithmetic sequence,

$r_i=0+\dfrac i2=\dfrac i2$

with $1\le i\le6$.

We approximate the integral of $f(x)$ on the interval [0, 3] by the Riemann sum,

$\displaystyle\int_0^3f(x)\,\mathrm dx=\sum_{i=1}^6f(r_i)\Delta x_i$

$\displaystyle=\frac{3-0}6\sum_{i=1}^6\left(3{r_i}^2-8r_i\right)$

$\displaystyle=\frac12\sum_{i=1}^6\left(\frac{3i^2}4-4i\right)$

$\displaystyle=\frac38\sum_{i=1}^6i^2-2\sum_{i=1}^6i$

Recall the formulas,

$\displaystyle\sum_{i=1}^ni=\frac{n(n+1)}2$

$\displaystyle\sum_{i=1}^ni^2=\frac{n(n+1)(2n+1)}6$

Then the value of the integral is approximately

$\displaystyle=\frac38\cdot\frac{6\cdot7\cdot13}6-2\cdot\frac{6\cdot7}2=\boxed{-\frac{63}8}=-7.875$

Compare to the exact value of the integral, -9.