3.7 Definite integrals

1.
(a) $\displaystyle{\lim_{n \to \infty}\frac{1}{n}\sum_{i=1}^{n}\left(\frac{i}{n}\right)^{2} = \frac{1}{3}}$
2.
(a) $\displaystyle{\frac{10}{3}}$ (b) $\displaystyle{\frac{3}{2} - \log{2}}$ (c) $\displaystyle{\frac{2}{5}}$ (d) $\displaystyle{0}$ (e) $\displaystyle{1}$
3.
(a) $\displaystyle{5}$ (b) $\displaystyle{-2}$ (c) $\displaystyle{-1}$ (d) $\displaystyle{0}$ (e) $\displaystyle{-4}$
4.
(a) $\displaystyle{\sin{x}}$ (b) $\displaystyle{-\cos{x}}$ (c) $\displaystyle{2\sqrt{\sin{2x}}}$
5.
(a) $\displaystyle{\frac{1}{2}}$ (b) $\displaystyle{\log{\frac{3}{2}}}$ (c) $\displaystyle{\frac{2}{3}}$