Question

Evaluate the line integral in Stokes? Theorem to evaluate the surface integral integrate S ( x F) n dS. Assume that n is in the positive z-direction. F=(x+y,y+z,z+x); Sis the tilted disk enclosed by r(t)=(cost,4 sin t root 5 cost). Integrate s ( * F). n ds= (Type an exact answer. using pi as needed)

Answer 1

Answer:

Step-by-step explanation:

Consider that F(x,y,z) = (x + y, y + z, z + z)

$r(t)=(cos t,4sint,\sqt{5}cost)\\r'(t)=(-sint,4cost,-\sqrt{5}sint)\\F(r(t))=[tex]\int\limits^{2\pi}_0 {F(r(t))r'(t)} \, dt \\\\=\int\limits^{2\pi}_0(cost+4sint,4sint+\sqrt{5}cost+cost)(-sint,4cost,-\sqt{5}sint)dt\\\\=\int\limits^{2\pi}_0(-sintcost-4sin^2t+16costsint+4\sqrt{5}cos^2t-5sintcost-\sqrt{5}sintcost)dt\\\\=\int\limits^{2\pi}_0(10sintcost-4sin^2t+4\sqrt{5}cos^2t-\sqrt{5}sintcost)dt\\\\=\int\limits^{2\pi}_0((10-\sqrt{5})sintcost-(4+4+\sqrt{5})sin^2t+4\sqrt{5})dt\\\\(10-\qrt{5})(0)-(4+4+\sqrt{5}(\pi)+4\sqrt{5}(2\pi)\\\\(\sqrt{5}-1)4\pi)$[/tex]