Evaluate the following independent definite integrals on [0,1]. Show a suitable method.

1. Evaluate \(\displaystyle \int_{0}^{1} \frac{x}{x^{2}+ 49}\, dx\). Explain your reasoning. [2]
2. Evaluate \(\displaystyle \int_{0}^{1} (x^{2}+ 49)^{- \frac{1}{2}}\, dx\). Explain your reasoning. [2]
3. Evaluate \(\int_0^1 xe^{2x}\,dx\). Explain your reasoning. [2]

On \(0\le x\le 1\), the curve is \(\displaystyle y= \frac{2}{3}(x+ 7)^{\frac{3}{2}}\).

1. Find its arc length. Explain your reasoning. [2]
2. Find \(\displaystyle \frac{S}{\pi}\) for the curved surface produced by revolution about the x-axis. Explain your reasoning. [2]
3. Find \(\displaystyle \frac{V}{\pi}\) for the solid formed from the region under the curve. Explain your reasoning. [2]