(Solved): a) Express the limit as a definite integral on the given interval. (i) \( \lim _{n \rightarrow \inf ...

a) Express the limit as a definite integral on the given interval. (i) \( \lim _{n \rightarrow \infty} \sum_{k=1}^{n}\left(\left(x_{k}\right)^{3}-2 x_{k}\right) \ln \left(1+\left(x_{k}\right)^{2}\right) \Delta x, \quad[2,6] \) (ii) \( \lim _{n \rightarrow \infty} \sum_{k=1}^{n} \frac{\left(x_{k}\right)^{2} \cos x_{k}}{x_{k}+1} \Delta x, \quad[\pi, 2 \pi] \) b) Sketch the region enclosed by the given curves, then find the area of the region. \[ y=\cos x, \quad y=\sin (2 x), \quad x=0, \quad x=\pi \] c) Find the values of \( p \in \mathbb{R} \) for which the series is convergent. \[ \sum_{n=1} \frac{(-1)^{n-1}}{n^{p}} \]