Evaluate ʃ ʃ F۰dS if F=<−xy,3x^2,−yz> and the surface S is given by z=xsin(4y) for 0<=x<=pi/8 and 0<=y<=pi/8. (Take S to have upward orientation.)
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∫∫s F ۰ dS
= ∫∫ <-xy, 3x^2, -yz> ۰ <-z_x, -z_y, 1> dA, using cartesian coordinates
= ∫∫ <-xy, 3x^2, -xy sin(4y)> ۰ <-sin(4y), -4x cos(4y), 1> dA
= ∫(y = 0 to π/8) ∫(x = 0 to π/8) -12x^3 cos(4y) dx dy
= ∫(y = 0 to π/8) cos(4y) dy * ∫(x = 0 to π/8) -12x^3 dx
= [(1/4) sin(4y) {for y = 0 to π/8}] * [-3x^4 {for x = 0 to π/8}]
= (1/4) * -3(π/8)^4
= -3π⁴/16384.
I hope this helps!
= ∫∫ <-xy, 3x^2, -yz> ۰ <-z_x, -z_y, 1> dA, using cartesian coordinates
= ∫∫ <-xy, 3x^2, -xy sin(4y)> ۰ <-sin(4y), -4x cos(4y), 1> dA
= ∫(y = 0 to π/8) ∫(x = 0 to π/8) -12x^3 cos(4y) dx dy
= ∫(y = 0 to π/8) cos(4y) dy * ∫(x = 0 to π/8) -12x^3 dx
= [(1/4) sin(4y) {for y = 0 to π/8}] * [-3x^4 {for x = 0 to π/8}]
= (1/4) * -3(π/8)^4
= -3π⁴/16384.
I hope this helps!