Evaluate the work done P W = f F dr = f (Fxdx Fydy) 0 0

QUESTION:

Evaluate the work done

\(W=\int_{O}^{P} \mathbf{F} \cdot d \mathbf{r}=\int_{O}^{P}\left(F_{x} d x+F_{y} d y\right)\)

by the two-dimensional force \(\mathbf{F}=\left(x^{2}, 2 x y\right)\) along the three paths joining the origin to the point P = (1, 1) as shown in Figure 4.24(a) and defined as follows:

(a) This path goes along the x axis to Q = (1, 0) and then straight up to P. (Divide the integral into two pieces, \(\int_{O}^{P}=\int_{O}^{Q}+\int_{Q}^{P}\)).

(b) On this path \(y=x^{2}\), and you can replace the term dy in (4.100) by dy = 2x dx and convert the whole integral into an integral over x.

(c) This path is given parametrically as \(x=t^{3}, y=t^{2}\). In this case rewrite x, y, dx, and dy in (4.100) in terms of t and dt, and convert the integral into an integral over t.