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Pre-Calc Homework Solutions 431

# Pre-Calc Homework Solutions 431 - Section 10.6 22 25 431...

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22. [ 2 4.7, 4.7] by [ 2 3.1, 3.1] Use the symmetries of the graphs: the shared area is 4 E p /2 0 } 1 2 } [2(1 2 cos u )] 2 d u 5 8 E p /2 0 (1 2 2 cos u 1 cos 2 u ) d u 5 8 3 u 2 2 sin u 1 } 1 2 } u 1 } 1 4 } sin 2 u 4 5 6 p 2 16 23. For a 5 1: [ 2 3, 3] by [ 2 2, 2] The curves intersect at the origin and when 3 a cos u 5 a (1 1 cos u ) 2 cos u 5 1 u 5 6 } p 3 } . Use the symmetries of the curves: the area in question is 2 E p /3 0 } 1 2 } [(3 a cos u ) 2 2 a 2 (1 1 cos u ) 2 ] d u 5 a 2 E p /3 0 (9 cos 2 u 2 1 2 2 cos u 2 cos 2 u ) d u 5 a 2 E p /3 0 (8 cos 2 u 2 2 cos u 2 1) d u 5 a 2 3 4 u 1 2 sin 2 u 2 2 sin u 2 u 4 5 a 2 p 24. [ 2 3, 3] by [ 2 2, 2] The curves intersect when 6 cos 2 u 5 3 u 5 6 } p 6 } or 6 } 5 6 p } . Use the symmetries of the curves. The area in question is 4 E p /6 0 } 1 2 } (6 cos 2 u 2 3) d u 5 6 3 sin 2 u 2 u 4 5 3 ˇ 3 w 2 p . 25. [ 2 6, 6] by [ 2 4, 4] The area in question is half the circle minus two lobelike regions: } 1 2 } p (2) 2 2 2 E p /2 0 } 1 2 } [2(1 2 sin u )] 2 d u 5 2 p 2 E p /2 0 (4 2 8 sin u 1 4 sin 2 u ) d u 5 2 p 2 4 3 u 1 2 cos u 1 } 1 2 } u 2 } 1 4 } sin 2 u 4 5 8 2 p 26. [ 2 3, 3] by [ 2 2, 2] (a) To find the integration limits, solve
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