multivariable line integral

Question

Racing Superman from Gotham City to the Pacific Ocean, the Flash exerts the force F~ = 4xy~i+3~j and travels along the path x = y 2+4 from (4, 0) to (5, 1). By evaluating a line integral, determine how much work is done by the Flash

Answer 1

So we use parametric equations for the path. y = t, x = t² + 4. We can put this in a vector.

r(t) = t²+4 i + t j

Find F(r(t))

F(r(t)) = 4*(t² + 4) * t i + 3 j = 4t³ + 16t i + 3 j

Next we find r'(t) by taking derivatives of the component vectors of r(t) with respect to t

r'(t) = 2t i + 1 j

Plug it into the equation: ∫_C F • dr

F • dr is (4t³ + 16t)(2t) + 3(1) = 8t⁴ + 32t² + 3

Now get the interval:

y = t and y goes from 0 to 1. Therefore t goes from 0 to 1

Plug into the line integral equation and solve

∫₀¹ 8t⁴ + 32t² + 3 = 8/5 + 32/3 + 3 = 229/ 15