Split up the interval [0, 8] into 4 equally spaced subintervals:
[0, 2], [2, 4], [4, 6], [6, 8]
Take the right endpoints, which form the arithmetic sequence
[tex]r_i=2+\dfrac{8-0}4(i-1)=2i[/tex]
where 1 ≤ i ≤ 4.
Find the values of the function at these endpoints:
[tex]f(r_i)=-4{r_i}^2+32r_i=-16i^2+64i[/tex]
The area is given approximately by the Riemann sum,
[tex]\displaystyle\int_0^8f(x)\,\mathrm dx\approx\sum_{i=1}^4f(r_i)\Delta x_i[/tex]
where [tex]\Delta x_i=\frac{8-0}4=2[/tex]; so the area is approximately
[tex]\displaystyle2\sum_{i=1}^4(-16i^2+64i)=-32\sum_{i=1}^4i^2+128\sum_{i=1}^4i=-32\cdot\frac{4\cdot5\cdot9}6+128\cdot\frac{4\cdot5}2=\boxed{320}[/tex]
where we use the formulas,
[tex]\displaystyle\sum_{i=1}^ni=\frac{n(n+1)}2[/tex]
[tex]\displaystyle\sum_{i=1}^ni^2=\frac{n(n+1)(2n+1)}6[/tex]
