Proof (a) Let be the line segment joining and Show that (b) Let be the vertices of a

Chapter 15, Problem 47

(choose chapter or problem)

Proof

(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 \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)\) 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_1 y_2 - x_2 y_1

(x_1, y_1),(x_2, y_2), … ,(x_n, y_n)

1 / 2 [(x_1 y_2 - x_2 y_1) + (x_2 y_3 - x_3 y_2) + cdots +

(x_n - 1} y_n - x_n y_n - 1) + (x_n y_1 - x_1 y_n)]

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