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,\frac\pi4\right],\left[\frac\pi4,\frac\pi2\right],\ldots,\left[\frac{5\pi}4,\frac{3\pi}2\right]`

where the right endpoints are given by

`r_i=i\frac{\frac{3\pi}2-0}6=\frac{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(\frac1{\sqrt2}+1+\frac1{\sqrt2}+0-\frac1{\sqrt2}-1\right)=\frac\pi{\sqrt2}\approx\boxed{2.221441}`

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