Find the work done by the force field f in moving an object from a to b. f(x, y) = 2y3/2 i + 3x y j a(1, 1), b(3, 9)

To answer this question we need to apply the concept of work, that is

w = ∫ ₐᵇ F×ds

Solution is:

w = 523 J

w = ∫ₐᵇ F × ds

F = 2y³/² i + 3xy j        and    ds = dx i + dy j

Then  F × ds  = ( 2y³/² i + 3xy j ) × ( dx i + dy j )

F × ds  = 2y³/² dx + 3xy dy

And     = ∫ₐᵇ F×ds   =  ∫ₐᵇ2y³/² dx + 3xy dy       a  ( 1 , 1 )   b ( 3 , 9 )

∫ₐᵇ F×ds   =  2y³/²∫dx   +  3x ∫ydy

∫ₐᵇ F×ds   =  2y³/²x  |₁,₁³,⁹  + 3xy²/2  |₁,₁³,⁹

∫ₐᵇ F×ds   = 2 × (9)³/² × (3) - 2 × (1 ) ³/²× (1) +  3 × (3)× (9)² /2 - 3 × (1)³/²/2

∫ₐᵇ F×ds   =  162 - 2 + 364,5 - 1,5

∫ₐᵇ F×ds   = 160 + 363

∫ₐᵇ F×ds   = 523 J

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go and ask your mom to you

the answer is d

the answer is in the first quadrant, which make the y-int positive.

since the dots are going down, therefore the correlation is negative

i hope it is right.