Question 27.3: Evaluate ∫C F.ds where F = 5y² i + 2xyj and C is the strai......

We compare the integrand with the standard form and recognize that in this case P(x, y)=5 y^2 \text { and } Q(x, y)=2 x y \text {. } The integral becomes

The curve of interest, in this case a straight line, is shown in Figure 27.5. Along this curve it is clear that y = x at all points. We use this fact to simplify the integral by writing everything in terms of x. We could equally well choose to write everything in terms of. If y=x \text { then } \frac{\mathrm{d} y}{\mathrm{~d} x}=1 \text {, that is } \mathrm{d} y=\mathrm{d} x \text {. } As we move along the curve C, x ranges from 0 to 1, and the integral reduces to