(a) Let be the line segment joining and Showthat(b) Let be the vertices of apolygon

Chapter 15, Problem 44

(choose chapter or problem)

(a) Let C be the line segment joining \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\). Show that \(\int_{C}-y d x+x d y=x_{1} y_{2}-x_{2} y_{1}\).

(b) Let \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)\), . . . , \(\left(x_{n}, y_{n}\right)\) be the vertices of a polygon. Prove that the area enclosed is

\(\frac{1}{2}\left[\left(x_{1} y_{2}-x_{2} y_{1}\right)+\left(x_{2} y_{3}-x_{3} y_{2}\right)+\cdots+\right. \left.\left(x_{n-1} y_{n}-x_{n} y_{n-1}\right)+\left(x_{n} y_{1}-x_{1} y_{n}\right)\right]\)

Text Transcription:

(x_1,y_1)

(x_2,y_2)

int_C -y dx + x dy = x_1y_2-x_2y_1

(x_n,y_n)

1/2[(x_1y_2-x_2y_1)+(x_2y_3-x_3y_2)+ cdots + (x_n-1 y_n -x_n y_n-1)+(x_ny_1-x_1y_n)]