How do you find the area ( if any ) common to the four cardioids $r = 1 \pm cos θ and r = 1 \pm sin θ$ $r=1+-cos theta and r = 1+-sin theta$ ?

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How do you find the area ( if any ) common to the four cardioids $r = 1 \pm cos θ and r = 1 \pm sin θ$ $r=1+-cos theta and r = 1+-sin theta$ ?

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Answer 1 · 2016-09-29T02:43:58

$\frac{3}{2} π - 4 \sqrt{2} + 1 = 0.055535$ $3/2pi-4 sqrt 2 +1=0.055535$ areal units.

Explanation:

The lines of symmetry (axes ) of the cardioids are the positive and

negative axes of coordinates.

The cardioids are equal in size and symmetrically placed, with

respect to the pole r=0 that is a common point of imtersection for all.

The common area comprises four equal parts.

The other terminal points, in these parts, are

$(1 - \frac{1}{\sqrt{2}}, \frac{π}{4}) \in Q_{1}$ $(1-1/sqrt 2, pi/4) in Q_1$

$(1 - \frac{1}{\sqrt{2}}, \frac{3}{4} π) \in Q_{2}$ $(1-1/sqrt 2, 3/4pi) in Q_2$

$(1 - \frac{1}{\sqrt{2}}, \frac{5}{4} π) \in Q_{3}$ $(1-1/sqrt 2, 5/4 pi) in Q_3$

$(1 - \frac{1}{\sqrt{2}}, \frac{7}{4} π) \in Q_{4}$ $(1-1/sqrt 2, 7/4pi) in Q_4$

Now, one such part is the area in $Q_{3}$ $Q_3$ that is

symmetrical about $θ = \frac{5}{4} π$ $theta = 5/4pi$ and bounded by

$r = 1 + cos θ$ $r = 1+cos theta$ , between $(0, π) and (1 - \frac{1}{\sqrt{2}}, \frac{5}{4} π)$ $(0, pi) and (1-1/sqrt 2, 5/4pi)$ and

$r = 1 + sin θ$ $r = 1+sin theta$ , between $(0, \frac{3}{2} π) and (1 - \frac{1}{\sqrt{2}}, \frac{5}{4} π)$ $(0, 3/2pi) and (1-1/sqrt 2, 5/4pi)$

This area = $2 \int \int r d r d θ$ $2 int int r dr d theta$ ,

r from 0 to $(1 + cos θ) and θ$ $(1+cos theta) and theta$ from $π$ $pi$ t.o $\frac{5}{4} π$ $5/4pi$ .

After integration with respect to r, this becomes

$\int {(1 + cos θ)}^{2} d θ$ $int (1+cos theta)^2 d theta$ , between the limits $π and \frac{5}{4} π .$ $pi and 5/4pi.$

$= \int (1 + 2 cos θ + {cos}^{2} θ) d θ$ $= int (1+2 cos theta + cos^2theta) d theta$ , between the limits

$= \int (1 + 2 cos θ + \frac{1 + cos 2 θ}{2}) d θ$ $= int (1+2 cos theta + (1+ cos 2theta)/2) d theta$ , between the limits

$= [\frac{3}{2} θ + 2 sin θ + \frac{1}{4} sin 2 θ]$ $= [3/2 theta + 2 sin theta +1/4sin 2 theta]$ , between $π and \frac{5}{4} π .$ $pi and 5/4pi.$

$= \frac{3}{8} π - \frac{2}{\sqrt{2}} + \frac{1}{4}$ $= 3/8pi-2/sqrt 2+1/4$

$= \frac{3 π - 8 \sqrt{2} + 2}{8}$ $=(3pi-8 sqrt 2 + 2)/8$

The total common area is 4 X this area

$= \frac{3}{2} π - 4 \sqrt{2} + 1 = 0.055535$ $= 3/2pi-4 sqrt 2 +1=0.055535$ areal units

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How do you find the area ( if any ) common to the four cardioids $r = 1 \pm cos θ and r = 1 \pm sin θ$ $r=1+-cos theta and r = 1+-sin theta$ ?

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Explanation:

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Science

Math

Humanities

... and beyond

How do you find the area ( if any ) common to the four cardioids r=1±cosθandr=1±sinθr=1+-cos theta and r = 1+-sin theta?

1 Answer

Explanation:

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How do you find the area ( if any ) common to the four cardioids $r = 1 \pm cos θ and r = 1 \pm sin θ$ $r=1+-cos theta and r = 1+-sin theta$ ?