9. Evaluate / x + z ds where C is the semicircle x + y = 9 with z =5 and x 2 0. Assign the result to q9. 10. Evaluate / / * + zdS where E is the part of the cylinder x3+ z =9 between y = 1 and y = 3. Assign the result to q10. E 11. Evaluate / / (yi + x] + zk) . ndS where E is the part of the paraboloid z = 9 - x2 - y inside the cylinder r = 2 cos @ with downwards orientation. Assign the result to q11.Surface Integrals of Functions Ditto for surface integrals of functions. For example to calculate the surface integral / {, xydS where > is parametrizationy7 + (x2 + y?)k with 0 (1), z(1)|ru X TulldA E R Of course Matlab doesn't know how to do the latter integral so we need to make sure we iterate it appropriately. Here we parametrization y as the inner variable and x as the outer variable: syms x y z; rbar = [x, y, x^2+y^2]; f = x*y; int(int(simplify(subs(f, [x, y, z], rbar)*norm(cross(diff(rbar, x) , diff(rbar, y) ) ) ) , x, 0, 2) , y, 0, 3) This yields output: ans = (2809*53^(1/2) )/240 - (1369*37^(1/2) )/240 - (289*17^(1/2) )/240 + 1/240Surface Integrals of Vector Fields When dealing with surface integrals of vector fields we encounter the surface orientation issue. Our basic method of integration is: ( / F(x. >,2) . nas = = / F(x.y,2) . ( Fux TV) dA M. R Where the + indicates that we need to decide which it is. Ideally we'd like to parametrizationthat the cross-product generated normal vectors point in the direction of desired orientation but this is quite difficult. Instead we take the approach from class - we look at the cross product , X r, and examine whether it agrees with or disagrees with E's orientation. If it disagrees we negate. For example suppose we wish to evaluate / J (x27 + y] + zk) . ndS where > is the part of the parabolic sheet y = > with -2