Chapter 15.4, Problem 38ES

(a) Let C be the line segment from a point $\left(a,b\right)$ to a point $\left(c,d\right).$ Show that ${\displaystyle {\int }_C-ydx+xdy=ad-bc}$

(b) Use the result in part (a) to show that the area A of a triangle with successive vertices $\left(x_1,y_1\right),\left(x_2,y_2\right)\text{and}\left(x_3,y_3\right)$ going counterclockwise is $A=\frac{1}{2}\left[\left(x_1{y}_2-x_2{y}_1\right)\right]+\left(x_2{y}_3-x_3{y}_2\right)+\left.\left(x_3{y}_1-x_1{y}_3\right)\right]$

(c) Find a formula for the area of a polygon with successive vertices $\left(x_{1,}{y}_1\right),\left(x_2,y_2\right),\dots ,\left(x_n,y_n\right)$ going counterclockwise.

(d) Use the result in part (c) to find the area of a quadrilateral with vertices $\left(0,0\right),\left(3,4\right),\left(-2,2\right),\left(-1,0\right).$