a) Change the order of integration in \( \int_{0}^{0} \int_{0}^{3}\left(x^{2}+y^{2}\right) d y d x \) and hence evaluate it.

2. Determine the volume of the solid obtained by rotating the region bounded by \( y=\sqrt{x}, y=3 \) and the \( y \)-axis about the \( y \)-axis.

Determine if the given sequence converges or diverges. If it converges what is its limit? \[ \left\{\frac{n^{2}-7 n+3}{1+10 n-4 n^{2}}\right\}_{n=3}^{\infty} \]

(20p)Evaluate \( \left|\begin{array}{rrcrr}0 & 3 & 1 & -2 & 4 \\ 1 & 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & -1 & 0 \\ 3 & 2 & 0 & 10 & 0 \\ 4 & 0 & 0 & 5 & 1\end{array}\right| \)

(b) Prove that between any two real roots of \( e^{x} \cos 2 x=2 \), there is at least one real root of \( e^{x} \sin 2 x=1 \)

Find the exact area of the surface obtained by rotating about the \( x \)-axis the curve: \( x=\frac{1}{3}\left(y^{2}+2\right)^{\frac{3}{2}}, 1 \leq y \leq 2 \).