Now some enclosed integrals. These are line and surface integrals. A closed line integral is a path that ends where it begins. A closed surface is an area that wraps ahas no openings. Both are denoted by the circle in the middle of the integral, $o{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}ds$ and $dA($usually this is just denoted by $o{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}dA,$ so watch for the type of integrand$).$ These will be used in Maxwell's equations if the integrand can be simplified to a scalar.

$14.o{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}(1x{x}^2)ds=$ along the square path of length $5$ starting at $x=1$ and contained in the first quadrant. Remember x will be constant along the vertical paths.

$15.vec(P)*vec(dA)=,$ where vec$(P)=$rhat$(r)$ and vec$(dA)=\sqrt[\phantom{2}]{(cos({)}^{2}sin({)}^2)}sin()d\frac{O}{d}hat(r).$ Use limits of $\frac{O}{?}=0$ to $2,=0$ to $,r=0$ to $r.$