Question

The line integral ${\int }_{P_1}^{P_2}(ydx+xdy)$ from $P_1(x_1,y_1$) to $P_2(x_2,y_2$) along the semi-circle $P_1{P}_2$ shown in the figure is

• A. $x_2{y}_2-x_1{y}_1$
• B. $(y_2^2-y_1^2)+(x_2^2-x_1^2)$
• C. $(x_2-x_1$)$(y_2-y_1$)
• D. $(y_2-y_1{)}^2+(x_2-x_1{)}^2$

Answer 1

The correct option is A $x_2{y}_2-x_1{y}_1$

I = ${\int }_{P_1}^{P_2}(ydx+xdy)$
= ${\int }_{P_1}^{P_2}d\left(xy\right)=[xy{]}_{P_1}^{P_2}$
$=x_2{y}_2-x_1{y}_1$