Question

(a) Find the Riemann sum for f(x) = 4 sin(x), 0 ≤ x ≤ 3π/2, with six terms, taking the sample points to be right endpoints. (Round your answers to six decimal places.)

Answer 1

Split up the integration interval into 6 subintervals:

$\left[0,\dfrac\pi4\right],\left[\dfrac\pi4,\dfrac\pi2\right],\ldots,\left[\dfrac{5\pi}4,\dfrac{3\pi}2\right]$

where the right endpoints are given by

$r_i=i\dfrac{\frac{3\pi}2-0}6=\dfrac{i\pi}4$

for $1\le i\le6$. Then we approximate the integral

$\displaystyle\int_0^{3\pi/2}4\sin x\,\mathrm dx$

by the Riemann sum,

$\displaystyle\sum_{i=1}^6f(r_i)\frac{\frac{3\pi}2-0}6=\pi\sum_{i=1}^6\sin\frac{i\pi}4$

$=\pi\left(\dfrac1{\sqrt2}+1+\dfrac1{\sqrt2}+0-\dfrac1{\sqrt2}-1\right)=\dfrac\pi{\sqrt2}\approx\boxed{2.221441}$

Compare to the actual value of the integral, which is exactly 4.