AREAS AND LENGTHS IN POLAR COORDINATES. 25. Find the area inside the larger loop and outside the smaller loop

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1 SECTIN 9. AREAS AND LENGTHS IN PLAR CRDINATES 9. AREAS AND LENGTHS IN PLAR CRDINATES A Click here for answers. S Click here for solutions. 8 Find the area of the region that is bounded by the given curve and lies in the specified sector.. r,. r e,. r cos,. r,. r sin, 6. r cos, 7. r sin, 8. r, Sketch the curve and find the area that it encloses. 9. r sin. r sin. r sin. r cos. r cos. r sin. r cos 6. r sin ; 7. Graph the curve r cos 6 and find the area that it encloses. ; 8. The curve with polar equation r sin cos is called a bifolium. Graph it and find the area that it encloses. 9 Find the area of the region enclosed by one loop of the curve. 9. r cos. r sin. r sin. r cos inner loop) Find the area of the region that lies inside the first curve and outside the second curve.. r cos,. r cos,. Find the area inside the larger loop and outside the smaller loop of the limaçon r sin. ; 6. Graph the hippopede r s.8 sin and the circle r sin and find the exact area of the region that lies inside both curves. 7 Find the length of the polar curve. 7. r cos, 8. r, 9. r cos. r e,.. r cos r cos r cos r Use a calculator or computer to find the length of the loop correct to four decimal places.. ne loop of the four-leaved rose r cos. The loop of the conchoid r sec Copyright, Cengage Learning. All rights reserved.

2 SECTIN 9. AREAS AND LENGTHS IN PLAR CRDINATES 9. ANSWERS E Click here for exercises. S Click here for solutions.. 6. e e ) ) ) ). sin +9 7 ). 7 cos arcsin., ). +ln ) ln e ) , 6. Copyright, Cengage Learning. All rights reserved., 9,

3 SECTIN 9. AREAS AND LENGTHS IN PLAR CRDINATES 9. SLUTINS E Click here for exercises.. A r dθ θ dθ θ] A / / eθ dθ eθ] / / e e ). A /6 cos θ) dθ /6 + cos θ) dθ θ + sin θ] /6. A /6 /6. A /6 6 + /θ) dθ / θ)] /6 /6 sin θdθ /6 cos θ) dθ θ 6 sin θ] / A / cos θdθ / + cos 6θ) dθ θ + sin 6θ] / +) 6 7. A / / sin θ) dθ / 9 cos θ) dθ / 9 θ sin θ] / 9 +) / 8 8. A / / θ ) dθ θ] / / 6. A 6 /6 sin θdθ /6 cos 6θ) dθ θ sin 6θ] /6 6 A cos θ)] dθ 6 cosθ +cos θ ) dθ 8 cosθ +cosθ) dθ 6θ 8sinθ + sin θ] 9.., ). A sin θ) dθ θ sin θ] cos θ) dθ A / cos θ) dθ / + cos θ) dθ θ + sin θ] /. Copyright, Cengage Learning. All rights reserved. A / / sin θ) dθ / / 6 8sinθ +sin θ ) dθ / / 6 + sin θ ) dθ by Theorem..7b)] / / 6 + sin θ ) dθ by Theorem..7a)] 6 + cos θ)] dθ θ sin θ] / A / + sin / θ) dθ / / + sin θ + sin θ ) dθ θ cosθ] / / + / cos θ) dθ / + θ sin θ] / + /

4 SECTIN 9. AREAS AND LENGTHS IN PLAR CRDINATES.,,, 9. A /6 cos θdθ /6 + cos 6θ) dθ θ + sin 6θ] / A cos θ) dθ 9 6cosθ +cos θ ) dθ 9θ 6sinθ + θ + sin θ] 9. A / sin θ) dθ 9 / cos θ) dθ 9 θ sin θ] / 9 8. A / sin θdθ θ sin θ] / / cos θ) dθ. 7. A 8 / sin θdθ / cos 8θ) dθ θ sin 8θ] / By symmetry, the total area is twice the area enclosed above the polar axis, so A r dθ + cos 6θ] dθ +cos6θ +cos 6θ ) dθ θ + sin 6θ) + sin θ + θ)] cosθ cos θ θ cos ) or cos ) ).Letα cos.then A + cos α θ) dθ α +cosθ +9cos θ ) dθ 7 α +cosθ + 9 cos θ) dθ 7 θ + sin θ + 9 sin θ] α 7 α) sin α 9 sin α cos α )] ) ) 7 cos ) 9 ) 7 cos 8. Copyright, Cengage Learning. All rights reserved. Note that the entire curve r sinθcos θ is generated by θ,]. The radius is positive on this interval, so the area enclosed is A r dθ sin θ cos θ ) dθ sin θ cos θdθ sin θ cos θ) cos θdθ sin θ) cos θdθ sin θ cos θ +)dθ sin θ cos θdθ+ sin θdθ ] θ sin θ] the first integral vanishes) 8 cos θ cos θ θ or A cos θ) ) ] dθ / / cosθ +cos θ ) dθ θ sinθ] / cos θ) dθ / θ + sin θ] /

5 SECTIN 9. AREAS AND LENGTHS IN PLAR CRDINATES Copyright, Cengage Learning. All rights reserved cosθ cos θ cos θ θ ± A / cos θ) cos θ) ] dθ / 8cos θ +cosθ ) dθ / cos θ +cosθ) dθ / sin θ + sin θ] The curve crosses itself when +sinθ sin θ.lettingα sin, the desired area is / A α + sin θ) dθ α + sin θ)] dθ / Now + sin θ) dθ 9θ cos θ +8θ sinθ + C,so A α +8cosα 6 sin α cos α The points of intersection occur where.8 sin θ sin θ.8sin θ θ arcsin α,socos α ). So the area is 9 A α. sin θdθ + / α.8 sin θ) dθ θ sin θ] α + θ.8 θ sin θ)] / α α sin α cos α)+.6.6α +. sin α cos α)] arcsin..6 7, +., r sin sin.6 arcsin. arcsin.. ) sin L b r a +dr/dθ) dθ / cos θ) + sin θ) dθ / cos θ + sin θdθ / dθ 8. L b a r +dr/dθ) dθ θ ) +ln) θ ] dθ +ln θ ln )] θ +ln dθ +ln ) ln 9. L + cos θ) + sin θ) dθ +cosθdθ cos θ/) dθ 8 sin θ/)] 8. L e θ ) + e θ ) dθ e θ dθ e ). L cos 8 θ) +cos 6 θ) sin θ) dθ cos θ) cos θ) + sin cos θ) dθ 8 / cos uduwhere u θ) θ) dθ 8 sin u sin u ] / 6 Note that the curve is retraced after every interval of length.. L cos θ)] + cos θ) sin θ)] dθ cos θ) dθ sin θ)]. From Figure in Example, L / r / +r ) dθ / cos θ + sin θdθ.6).. +secθ sec θ cos θ θ,. L / / + sec θ) + sec θ tan θ) dθ.88