Question

please solve all with detailed steps. thank you!

Find the mass, and the center of mass of the solid cone D with density p(x, y, z) = 1 bounded by the surface z = 4- x2 + y2 and z = 0 1) 2) Evaluate dA where R is the square with vertices (0,0), (1,–1), (2,0), and (1,1) x+y+1 (Hint: use a convenient change of variables) 3) Evaluate the line integral (x - y+ 2z)ds where C is the circle F(t) = (1,3 sin t,3 cos t) 0sts 2n %3D 4) Determine if the following vector field is conservative. If so, determine a potential function F(x.y.z) = (y+x, x+ z,x + y) 5) Find the circulation of the vector ficld F (x, y, z) = (z, x, -y)on the curve C: T(t) = (cost,sint,cos t) 0sts 2n 6) Consider the vector field F (x, y) = (y?, x2) on the region on the half annulus R = {(x, y): 1sx² + y? s9.y 2 0) whose boundary is C. Find the circulation on C assuming it has the orientation shown in figure. (Hint: use Green's theorem) %3D 7) Consider the vector field F (x, y, z) = (x, y, z) find the flux of F across the paraboloid z = 4-x - y? for z 20 8) Use Stoke's Theorem to evaluate the line integral F-di Where F=zt- zj + (x + y?)k and C consists of the three line segments that bound the plane z =8-4x -2y in the first octant oriented as shown in fig

Answer 1

037 let f (x, y, z) = a-y+22 » F û a scalar field. By definition of line integral of scalar field, f. F. dr = f(r(t)) . 11r' (t) Ildt here, rit) = < 1, 3 sint, 3 cost), ostsan. I r'lt) = 0, 3 cost, -3 sinty 11rLt) 1 = 1 0²+ ( 3cost)² + (-3sint) 118 (t) || = g cos²t +9 sin²t = 3 Fr(t) = 1-3sint +6 cost From above formula, { CX-y + 2z) ds = 3.5 (1-3 sint +6cost) dt. = 3. [ + + 3 cost tosint] 2 = 3 [ (271 + 3 COS(27) +6 sin (27)) - (0+3.40160 tosin.com] -3[ (271 +3 +o) - (3)] 67