evaluate the integral (F*dr) where F=3x^2i+(2xz-y)j+zk over an interval of C where C is the path given by x=t^2, y=t, z=2t^2-t, t is from 0 to 1 [0,1]
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∫c F · dr
= ∫c <3x^2, 2xz - y, z> · dr
= ∫(t = 0 to 1) <3t^4, 4t^4 - 2t^3 - t, 2t^2 - t> · <2t, 1, 4t - 1> dt
= ∫(t = 0 to 1) [6t^5 + (4t^4 - 2t^3 - t) + (2t^2 - t)(4t - 1)] dt
= ∫(t = 0 to 1) (6t^5 + 4t^4 + 6t^3 - 6t^2) dt
= 13/10.
I hope this helps!
= ∫c <3x^2, 2xz - y, z> · dr
= ∫(t = 0 to 1) <3t^4, 4t^4 - 2t^3 - t, 2t^2 - t> · <2t, 1, 4t - 1> dt
= ∫(t = 0 to 1) [6t^5 + (4t^4 - 2t^3 - t) + (2t^2 - t)(4t - 1)] dt
= ∫(t = 0 to 1) (6t^5 + 4t^4 + 6t^3 - 6t^2) dt
= 13/10.
I hope this helps!