Example 2: Work Done by a Force Vector Along a Curve

EXAMPLE 2

Let the force vector F be

F = -yi + xj.

F is perpendicular to the position vector xi + yj but has the same length as xi + yj. Find the work done by F along the following curves, shown in Figure 13.2.8, from (0, 0) to (1, 1):

(a)    C₁: The line y = x, 0 ≤ x ≤ 1.

(b)    C₂: The parabola y = x²,0 ≤ x ≤ 1.

(c)     C₃: The curve, 0 ≤ x ≤ 1.

Figure 13.2.8

(a)    Put x = t, y = t.

The work is zero because the force F is perpendicular to dS along C₁.

(b)    Put x = t,y = t².

(c)     Put x = t³, y = t.