Answering a student's email questions

For question 6, you should set $x=0$ and solve for $t$. You'll get two values of $t$, but only the one that gives you $y\geq 3$ will work. Then set $y=3$ and solve for $t$. Again you'll get two values of $t$, but only one gives you $x\geq 0$.

For question 7, the furthest distance from the origin is when $r$ is greatest. Thus we are trying to maximize $r$. Then from one variable calculus, just take the derivative with respect to $\theta$ to find critical points. $\frac{dr}{d\theta}=0$. There's a geometric interpretation to this, but it's much easier to explain through pictures than through words. If requested, I'll explain this.

Question 8 is asking for when $r=0$. No derivatives required. Let me know if I'm wrong on this.

For question 10, your mistake might have been that the integral should go from $0$ to $\pi$ and not from $0$ to $2\pi$.