Evaluate the line integral integral_C (x + y)ds where C is the curve parametrize

Question

Evaluate the line integral integral_C (x + y)ds where C is the curve parametrized by x(t) = 3t, y(t) = 4t where 0 lessthanorequalto t lessthanorequalto 5. Find the gradient of the function f(x, y) = 2xy + sin x. For the vector field F(x, y) = (2y + cos x, 2x), compute the line integral integral_C F middot dr for each of the curves shown below. Is the vector field F above conservative? Explain why or why not. Find a potential function phi (x, y, z) for the vector field F = 2xi + cos yj + 4zk. Using the potential function, evaluate the line integral integral^(1, pi/2, 4)_(0, 0, 2) 2x dx + cos y dy + 4z dz.

Explanation / Answer

1)

1.a

2.c

3.b

2) x =3t =>dx/dt =3

y =4t=>dy/dt =4

ds=[(dx/dt)2+(dy/dt)2] dt

ds=[(3)2+(4)2] dt

ds =5 dt

c(x+y) ds

= [0 to 5](3t+4t) 5dt

= [0 to 5]35t dt

=[0 to 5](35/2)t2

=(35/2)52 -(35/2)02

=437.5

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