Answer to Question #260090 in Calculus for tarie

Question

) Find the value of integral Z C F · dr, where C is a semicircle parameterized by vecsr(t) = cost,sin t, 0 ≤ t ≤ π, and F =< −y, x > . 

Answer 1

Given, F=<-y,x>=-y i+x j

and r=<cost, sint>=cost i+ sint j.

By using the formula,

$\int_C F.dr=\int_a^b F(r(t)).r'(t) dt\\ =\int_0^\pi F(cost, sint) .(-sint \space i+ cost \space j) dt\\ =\int_0^\pi (-sint \space i+cost \space j) .(-sint \space i+ cost \space j) dt\\ =\int_0^\pi (sin^2t+cos^2t)dt\\ =\int_0^\pi dt\\ =[t]_0^\pi\\ =\pi$