Question

Find the work done by the force F=(x2 , - x y) in moving a particle from (0,0) to (1,2) along the parabola y=(2 x2) from (0,0) to (1,2) .

Answer 1

Answer on Question 42373, Math, Integral Calculus

$$\vec{F} = (x^2, -x, y)$$

Let us write parametric equation for parabola. $x = t; y = 2t^2$. Hence, one needs to find integral $\int \vec{F}(x(t), y(t)) d\vec{s}$ for $t = 0..1$, $d\vec{s} = (dx, dy) = (1, 4t)dt$.

$$\int \vec{F} \, d\vec{s} = \int_0^1 (t^2 - 4t \cdot 2t^3) \, dt = \int_0^1 (t^2 - 8t^4) \, dt = \left(\frac{t^3}{3} - 8\frac{t^5}{5}\right)\big|_0^1 = \frac{1}{3} - \frac{8}{5} = \frac{-19}{15}.$$

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