Question

Consider points P and Q in the x-y plane, with P =(1,0) and Q = (0,1).The line integral 2${\int }_P^Q$(xdx+ydy) along the semicircle with the line segment PQ as its diameter

• A. is -1
• B. is 0
• C. is 1
• D. depends on the direction (clockwise or anitclockwise) of the semicircle)

Answer 1

The correct option is B is 0

I = $2{\int }_P^{Q}xdx+ydy$
=$2{\int }_1^{0}xdx+2{\int }_0^{1}ydy$
=$2{\left[\frac{x^2}{2}\right]}_1^0+2{\left[\frac{y^2}{2}\right]}_0^1$
=-1 +1
= 0