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

(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: \( x=\frac{1}{3}\left(y^{2}+2\right)^{\frac{3}{2}}, 1 \leq y \leq 2 \).

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

Find the exact area of the resulting surface obtained by rotating about the y-axis the curve: \( y=\frac{1}{3} x^{\frac{3}{2}}, 0 \leq x \leq 12 \).

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