Compute the upper and lower Riemann sums and the integral from zero to 1 of f(x)dx, where the partition is
P= {0, 2/5, 1/2, 3/5, 1}
Thanks.
$\displaystyle f(x) = 1 - x^2$
f(x) is decreasing from 0 to 1. hence, for the given partition:
$\displaystyle U(P,f) = 1\cdot \frac{2}{5} + \frac{21}{25} \cdot \frac{1}{10} + \frac{3}{4} \cdot \frac{1}{10} + \frac{16}{25} \cdot \frac{2}{5} = \frac{163}{200}$
$\displaystyle L(P,f) = \frac{21}{25} \cdot \frac{2}{5} + \frac{3}{4} \cdot \frac{1}{10} + \frac{16}{25} \cdot \frac{1}{10} + 0 \cdot \frac{2}{5} = \frac{19}{40}$