91Ó°ÊÓ

Evaluate the line integral $$ \int_{C} \boldsymbol{F} \cdot \boldsymbol{d} \boldsymbol{r} \quad \text { where } \quad \boldsymbol{F}=\left(5 z^{2}, 2 x, x+2 y\right) $$ and the curve \(C\) is given by \(x=t, y=t^{2}, z=t^{2}, 0 \leq t \leq 1\).

The surface \(S\) is defined to be that part of the plane \(z=0\) lying between the curves \(y=x^{2}\) and \(x=y^{2} .\) Find the surface integral of \(\boldsymbol{u} \cdot \boldsymbol{n}\) over \(S\) where \(\boldsymbol{u}=\left(z, x y, x^{2}\right)\) and \(\boldsymbol{n}=(0,0,1)\).

Find the volume of the section of the cylinder \(x^{2}+y^{2}=1\) that lies between the planes \(z=x+1\) and \(z=-x-1\).