(a) If C is the line segment connecting the point (x1, y1) to the point (x2, y2), show that
∫C x dy − y dx = x1y2 – x2y1
(b) If the vertices of a polygon, in counterclockwise order, are (x1, y1), (x2, y2),
, (xn, yn), show that the area of the polygon is
[(x1, y2), (x2, y1) + (x2y3 – x3y2) +
⋯ + (x
– xn yn−1) + (xny1 –
(c) Find the area of the pentagon with vertices (0, 0), (2, 1), (1, 3), (0, 2), and (−1, 1).
To show: For line segment connecting point to point , .
Points are and .
Write the expression for line segment which starts at vector and terminates at vector .
t is parameter.
Write the parametric equations of line segment C, with start point and terminal point by the use of equation (1).
Differentiate equation (2) with respect to t.
Differentiate equation (3) with respect to t.
Find the value of
To show: The area of polygon with vertices as .
To find: The area of pentagon with vertices , , , , and .
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