Chapter 16.4, Problem 21E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

(a) If *C* is the line segment connecting the point (*x*_{1}, *y*_{1}) to the point (*x*_{2}, *y*_{2}), show that

∫* _{C} x dy* −

(b) If the vertices of a polygon, in counterclockwise order, are (*x*_{1}, *y*_{1}), (*x*_{2}, *y*_{2}), `⋯ `

`,`

(*x _{n}*,

*A* =
*x*_{1}, *y*_{2}), (*x*_{2}, *y*_{1}) + (*x*_{2}*y*_{3} – *x*_{3}*y*_{2}) + `⋯ + (`

*x*_{n}_{−1}*y _{n}*

` – `*x*_{n} *y*_{n}_{−1}) + (*x*_{n}y_{1} –

*x*

_{1}

*y*_{n}

`)]`

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

**(a)**

To determine

**To show**: For line segment connecting point

Explanation

**Given data:**

Points are

**Formula used:**

Write the expression for line segment which starts at vector

Here,

*t* is parameter.

Write the parametric equations of line segment *C*,

Differentiate equation (2) with respect to *t*.

Differentiate equation (3) with respect to *t*.

Find the value of

**(b)**

To determine

**To show**: The area of polygon with vertices

**(c)**

To determine

**To find**: The area of pentagon with vertices

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