Question 8 (20 points) Parts (a) and (b) are distinct from e…

Questiоn 8 (20 pоints) Pаrts (а) аnd (b) are distinct frоm each other. (a)  Use Stokes' Theorem to evaluate (displaystyle iint_S text{curl} vec{G} cdot dvec{S}), where (vec{G}(x,y,z)= langle yz, xy, xz rangle) and (S) is the part of the sphere (x^2+y^2+z^2=4) with (x geq 0) oriented in the direction of the positive x-axis that lies inside the cylinder (y^2+z^2=1). (b) Consider the solid region E bounded below by the cone (z=sqrt{x^2+y^2}) and above by the paraboloid (z=2-(x^2+y^2)). Use the Divergence Theorem and cylindrical coordinates to evaluate (displaystyle iint_S vec{F} cdot dvec{S}), where (vec{F} (x,y,z)= (5xsin^2(z)+arctan(y))vec{i}+(5ycos^2(z)-ln(1+x^2))vec{j}+(7z-e^{x-y})vec{k}), and S is the boundary of the region E.