Question

Find the approximate area between the curve f(x) = -4x² + 32x and on the x-axis on the interval [0,8] using 4 rectangles. Use the right endpoint of each rectangle to determine the height.

Answer 1

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

$r_i=2+\dfrac{8-0}4(i-1)=2i$

where 1 ≤ i ≤ 4.

Find the values of the function at these endpoints:

$f(r_i)=-4{r_i}^2+32r_i=-16i^2+64i$

The area is given approximately by the Riemann sum,

$\displaystyle\int_0^8f(x)\,\mathrm dx\approx\sum_{i=1}^4f(r_i)\Delta x_i$

where $\Delta x_i=\frac{8-0}4=2$; so the area is approximately

$\displaystyle2\sum_{i=1}^4(-16i^2+64i)=-32\sum_{i=1}^4i^2+128\sum_{i=1}^4i=-32\cdot\frac{4\cdot5\cdot9}6+128\cdot\frac{4\cdot5}2=\boxed{320}$

where we use the formulas,

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

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