Evaluate ʃ ʃ F۰dS if F=<−6xy, 6x^2,−6yz> and the surface S is given by z = xe^y for 0<=x<=2 and 0<=y<=2. (Take S to have upward orientation.)
10 points for the best explanation.
Thanks.
Z
10 points for the best explanation.
Thanks.
Z
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∫∫s F۰dS
= ∫∫ <-6xy, 6x^2, -6yz> ۰ <-z_x, -z_y, 1> dA
= ∫(x = 0 to 2) ∫(y = 0 to 2) <-6xy, 6x^2, -6y*xe^y> ۰ <-e^y, -xe^y, 1> dy dx
= ∫(x = 0 to 2) ∫(y = 0 to 2) -6x^3 e^y dy dx
= ∫(x = 0 to 2) -6x^3 dx * ∫(y = 0 to 2) e^y dy
= (-3/2)x^4 {for x = 0 to 2} * e^y {for y = 0 to 2}
= -24(e^2 - 1).
I hope this helps!
= ∫∫ <-6xy, 6x^2, -6yz> ۰ <-z_x, -z_y, 1> dA
= ∫(x = 0 to 2) ∫(y = 0 to 2) <-6xy, 6x^2, -6y*xe^y> ۰ <-e^y, -xe^y, 1> dy dx
= ∫(x = 0 to 2) ∫(y = 0 to 2) -6x^3 e^y dy dx
= ∫(x = 0 to 2) -6x^3 dx * ∫(y = 0 to 2) e^y dy
= (-3/2)x^4 {for x = 0 to 2} * e^y {for y = 0 to 2}
= -24(e^2 - 1).
I hope this helps!
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right ,it is good