1.) $\displaystyle\int_{-10}^{10} \sqrt{4 \cos^2(t) + 25 + 4 \sin^2(t)} dt = \displaystyle\int_{-10}^{10} \sqrt{4 + 25} = 20 \sqrt{29}$

2.) $\vec{r}'(t) = <- \sin t, \cos t, 22t >$, so we have $|\vec{r}(t)| = \sqrt{1 + 484t^2}$ $$\vec{T}(t) = \frac{\vec{r}'(t)}{|\vec{r}'(t)|} = \left< -\frac{\sin t}{\sqrt{1+484t^2}}, \frac{\cos t}{\sqrt{1+484t^2}}, \frac{22t}{\sqrt{1+484t^2}} \right> $$ We also want the unit normal, $\vec{N}(t) = \frac{\vec{T}'(t)}{|\vec{T}'(t)|}$, so we compute $$\vec{T}'(t) = \left< -\frac{484 t^2 \cos(t) - 484 t \sin(t) + \cos(t)}{(484t^2+1)^{\frac{3}{2}}}, -\frac{484t^2 \sin(t) + \sin(t) + 484 t \cos(t)}{(484 t^2+1)^{\frac{3}{2}}}, \frac{22}{(484 t^2+1)^{\frac{3}{2}}} \right>,$$ and $$|\vec{T}'(t)| = \frac{1}{(484t^2 + 1)^{\frac{3}{2}}} \sqrt{ (484t^2 \cos (t) - 484t \sin(t) + \cos(t))^2 + (484t^2 \sin(t) + \sin(t) + 484t \cos(t))^2 + 484}$$ This is of course nasty, and we'd like to simplify it. You should expand the binomials and do some trig to simplify it. In the end simplification reduces to this: $$|\vec{T}'(t)| = \frac{1}{(484t^2+1)^{\frac{3}{2}}} \sqrt{234256t^4 + 235224t^2 + 485}$$ After this point, the rest of the calculations are relatively easy.
3.) You get $$\vec{T}(t) = \left< -\frac{\sin t}{\sqrt{1+ \tan^2(t)}}, \frac{\cos t}{\sqrt{1+\tan^2(t)}}, -\frac{\tan t}{\sqrt{1+\tan^2 t}} \right>$$ $$\vec{T}'(t) = \left< \cos(t) \cos(2t) \sqrt{\sec ^2(t)}, -\frac{2 \tan(t) \sec(t)}{(\sec^2(t))^{\frac{3}{2}}}, \frac{1}{\sqrt{\sec^2(t)}} \right>$$

It just gets nastier from here...compute $|\vec{T}'(t)|$ and then form $\vec{N}(t) = \frac{\vec{T}'(t)}{|\vec{T}'(t)|}$ (this will be some terrible formula). Then, compute $\vec{B}(t) = \vec{T}(t) \times \vec{N}(t)$, which will be another nasty computation.

4.) Write $\vec{r}(t) = < 2 \sin(3t), t, 2\cos(3t) >$, find $\vec{T}(t)$ and $\vec{N}(t)$ and $\vec{B}(t)$. To find the normal plane, which two vectors do you use? Which normal vector? To find the osculating plane, which two vectors do you use? Which normal vector?

5.) What's the normal vector of the given plane? Find an equation of the normal plane -- this equation should have $t$'s in it. Which $t$ do you pick to make the normal vector of the plane equal to the normal vector of the given plane?