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)

Step-by-step explanation:

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

" alt="\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)" />" />