(906, #21) (a) Evaluate the line integral

where F = < e^{x - 1}, x y> and C is given by r(t) = < t², t³>, t = 0 .. 1.

(b) Illustrate part (a) by using a graphing calculator or computer to graph C and the vectors from the vector field corresponding to t = 0, t = 1/sqrt(2), and t = 1

Solution:

a)

b) The graph of F (in red) on r shows that each component of F is increasing as t ranges from 0 to 1. Note that since F froms an acute angle with the tangent to the path, the line integral has a positive value. This confirms, in a rough way, the calculation of part (a).

by C.A.H.