H.W)Let \( f(x, y)=x y, \quad C: \vec{r}_{C}(t)=(t) i+(2 t) j, 0 \leq t \leq 1 \), evalaute \( I_{L, x C} \) and \( I_{L, x, C} \) Answer \( =\frac{2}{3},-\frac{2}{3} \)

Find \( \frac{d^{2} y}{d x^{2}} \) given that \( x^{3}-3 x y+y^{3}=1 \)

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

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 \( e^{x} \sin 2 x=1 \).

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.