Explanation Explanation Given 1. Area of integration : region bounded by the lemniscate curve with equation r 2 = sin ( 2 θ ) . Formula 1. Equation to transform Cartesian coordinates to polar coordinates 2. r d r d θ x = r cos ( θ ) , y = r sin ( θ ) r d r d θ 2. Area element in polar coordinates . Calculation Transform the integral into polar coordinates: x = r cos ( θ ) , y = r sin ( θ ) . Note that for the lemniscate, given a polar angle θ , r varies from zero to sin ( 2 θ ) , where θ varies from 0 to π 2 (since, lemniscate is in the first quadrant). Therefore the given integral can be written as: ∬ D x x 2 + y 2 d x d y = ∫ θ = 0 π / 2 ∫ r = 0 sin ( 2 θ ) r cos ( θ ) × r × r d r d θ = ∫ θ = 0 π / 2 ∫ r = 0 sin ( 2 θ ) r 3 cos ( θ ) d r d θ = ∫ θ = 0 π / 2 ( ∫ r = 0 sin

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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System that uses coordinates to uniquely determine the position of points. The most common coordinate system is the Cartesian system, where points are given by distance along a horizontal x-axis and vertical y-axis from the origin. A polar coordinate system locates a point by its direction relative to a reference direction and its distance from a given point. In three dimensions, it leads to cylindrical and spherical coordinates.

Chapter 16.4, Problem 24E

To determine

To find

The area integral $\iint_{D} x \sqrt{x^{2} + y^{2}} d x d y$ where D is specified.

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Chapter 16.4, Problem 23E

Chapter 16.4, Problem 25E

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Chapter 16 Solutions

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To find The area integral ∬ D x x 2 + y 2 d x d y where D is specified.

Solution Summary: The author explains the equation to transform Cartesian coordinates to polar coordinate. The integral can be written as follows.

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To find

The area integral $\iint_{D} x \sqrt{x^{2} + y^{2}} d x d y$ where D is specified.

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Chapter 16 Solutions

To find The area integral ∬ D x x 2 + y 2 d x d y where D is specified.

Solution Summary: The author explains the equation to transform Cartesian coordinates to polar coordinate. The integral can be written as follows.

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Chapter 16 Solutions

To find

The area integral $\iint_{D} x \sqrt{x^{2} + y^{2}} d x d y$ where D is specified.