Split up the integration interval into 6 subintervals:
[tex]\left[0,\dfrac\pi4\right],\left[\dfrac\pi4,\dfrac\pi2\right],\ldots,\left[\dfrac{5\pi}4,\dfrac{3\pi}2\right][/tex]
where the right endpoints are given by
[tex]r_i=i\dfrac{\frac{3\pi}2-0}6=\dfrac{i\pi}4[/tex]
for [tex]1\le i\le6[/tex]. Then we approximate the integral
[tex]\displaystyle\int_0^{3\pi/2}4\sin x\,\mathrm dx[/tex]
by the Riemann sum,
[tex]\displaystyle\sum_{i=1}^6f(r_i)\frac{\frac{3\pi}2-0}6=\pi\sum_{i=1}^6\sin\frac{i\pi}4[/tex]
[tex]=\pi\left(\dfrac1{\sqrt2}+1+\dfrac1{\sqrt2}+0-\dfrac1{\sqrt2}-1\right)=\dfrac\pi{\sqrt2}\approx\boxed{2.221441}[/tex]
Compare to the actual value of the integral, which is exactly 4.
