(a) Set up a double integral for calculating the flux of the vector field F(x, y, z) = xi+yj through the open-ended circular cylinder of radius 5 and height 8 with its base on the xy- plane and centered about the positive z-axis, oriented away from the z-axis. If necessary, enter as theta. B D - 1.₁² Flux = A = B = C = D = dz do (b) Evaluate the integral. JSF. Flux = F.dA=

Use the equation giving the flux of the vector field across the curve to calculate the flux of x + 1 y lã (x + 1)² + y²' (x + 1)² + y² F(x, y) = across C, the segment 7 ≤ y ≤ 9 along the y-axis, oriented upwards. (Use symbolic notation and fractions where needed.) I F. dr =

For the vector field G = (yey + 4 cos(4x + y))i + (xeªy + cos(4x + y)) ♬ find the line integral of G along the curve C from the origin along the x-axis to the point (4,0) and then counterclockwise around the circumference of the circle x² + y² = 16 to the point (4/√√2,4/√√2). ScĞ·dr = 0 }

Let f(x, y) = 7x cos (y). Find the conservative vector field F, which is the gradient of f. (Give your answer using component form or standard basis vectors. Express numbers in exact form. Use symbolic notation and fractions where needed.) F = (7 cos (y).-7x sin(y) Evaluate the line integral of F over the upper half of the unit circle centered at the origin, oriented clockwise. (Give an exact answer. Use symbolic notation and fractions where needed.) C F. dr = Incorrect 12 Question Source: Rogawski 4e Calculus Early Transcendentals Publisher: W.H. Freema 5.02. D