?(a) If C is the line segment connecting the point \(\left(x_{1}, y_{1}\right)\) to the

Chapter 14, Problem 25

(choose chapter or problem)

(a) If C is the line segment connecting the point \(\left(x_{1}, y_{1}\right)\) to the point \(\left(x_{2}, y_{2}\right)\), show that

\(\int_{C} x d y-y d x=x_{1} y_{2}-x_{2} y_{1}\)

(b) If the vertices of a polygon, in counterclockwise order, are \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)\), show that the area of the polygon is

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

\(\left.+\left(x_{n-1} y_{n}-x_{n} y_{n-1}\right)+\left(x_{n} y_{1}-x_{1} y_{n}\right)\right]\)

(c) Find the area of the pentagon with vertices (0,0),(2,1), (1,3),(0,2), and (-1,1)

Equation Transcription:

∫

Text Transcription:

(x_1, y_1)

(x_2, y_2)

Integral_C x dy - y dx = x_1 y_2 - x_2 y_1

(x_1, y_1), (x_2, y_2), ... , (x_n, y_n)

A = 1/2 [(x_1 y_2-x_2 y_1) + (x_2 y_3-x_3 y_2) + times times times

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