6. Observe the following PacMan. Recall that the $ x $ and $ y $ coordinates of any point on the unit circle are obtained by the relation $ (x, y) = (\cos(\theta), \sin(\theta)) $, where $ \theta $ is the angle between the ray connecting the point to the origin and the positive $ x $-axis.(a) Describe PacMan (the set $ P $) in polar coordinates.(b) Evaluate $ \int \int_P x \, dA $. Give your answer in exact form, not a decimal approximation.---**Diagram Explanation:**The diagram represents a circular sector resembling the character PacMan. The circle has a radius of 1 (unit circle). The sector labeled $ P $ is the region of the circle not included in the slice that forms the mouth of PacMan.In the diagram:- The circle is centered at the origin $(0, 0)$ in the $ xy $-coordinate system.- The positive $ x $-axis is indicated, with $ x = 1 $ and $ y = 0 $.- The radius forms an angle that splits the circle, with another line at $ x = \frac{-1}{\sqrt{2}} $, which likely defines the boundary of the PacMan's mouth. The main focus is on describing this sector in polar coordinates and evaluating the integral over the region $ P $.

a)Describe PacMan (the set P) in Polar coordinates. Let (x,y) denotes cartesian coordinates of the…

6. Observe the following PacMan. Recall that the x and y coordinates of any point on the unit circle are obtained by the relation (x, y) = (cos(0), sin(0)), where 0 is the angle between the ray connecting the point to the origin and the positive x-axis. (a) Describe PacMan (the set P) in polar coordinates. (b) Evaluate S Lx dA. Give your answer in exact form, not a decimal ap- proximation.

6. Observe the following PacMan. Recall that the x and y coordinates of any point on the unit circle are obtained by the relation (x, y) = (cos(0), sin(0)), where 0 is the angle between the ray connecting the point to the origin and the positive x-axis. (a) Describe PacMan (the set P) in polar coordinates. (b) Evaluate S Lx dA. Give your answer in exact form, not a decimal ap- proximation.

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

6. Observe the following PacMan. Recall that the $ x $ and $ y $ coordinates of any point on the unit circle are obtained by the relation $ (x, y) = (\cos(\theta), \sin(\theta)) $, where $ \theta $ is the angle between the ray connecting the point to the origin and the positive $ x $-axis.

(a) Describe PacMan (the set $ P $) in polar coordinates.

(b) Evaluate $ \int \int_P x \, dA $. Give your answer in exact form, not a decimal approximation.

---

**Diagram Explanation:**

The diagram represents a circular sector resembling the character PacMan. The circle has a radius of 1 (unit circle). The sector labeled $ P $ is the region of the circle not included in the slice that forms the mouth of PacMan.

In the diagram:

- The circle is centered at the origin $(0, 0)$ in the $ xy $-coordinate system.
- The positive $ x $-axis is indicated, with $ x = 1 $ and $ y = 0 $.
- The radius forms an angle that splits the circle, with another line at $ x = \frac{-1}{\sqrt{2}} $, which likely defines the boundary of the PacMan's mouth.

The main focus is on describing this sector in polar coordinates and evaluating the integral over the region $ P $.

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