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Pre-Calc Homework Solutions 299

# Pre-Calc Homework Solutions 299 - 2 E 0.8241(cos x 2 x 2 dx...

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37. First find the two areas. For the triangle, } 1 2 } (2 a )( a 2 ) 5 a 3 For the parabola, 2 E a 0 ( a 2 2 x 2 ) dx 5 2 3 a 2 x 2 } 1 3 } x 3 4 5 } 4 3 } a 3 The ratio, then, is 5 } 3 4 } , which remains constant as a approaches zero. 38. E b a [2 f ( x ) 2 f ( x )] dx 5 E b a f ( x ) dx , which we already know equals 4. 39. Neither; both integrals come out as zero because the 2 1-to-0 and 0-to-1 portions of the integrals cancel each other. 40. Sometimes true, namely when dA 5 [ f ( x ) 2 g ( x )] dx is always nonnegative. This happens when f ( x ) \$ g ( x ) over the entire interval. 41. [ 2 1.5, 1.5] by [ 2 1.5, 1.5] The curves intersect at x 5 0 and x 56 1. Use the area’s symmetry: 2 E 1 0 1 } x 2 2 1 x 1 } 2 x 3 2 dx 5 2 3 ln 1 x 2 1 1 2 2 } 1 4 } x 4 4 5 2 ln 2 2 } 1 2 } 5 ln 4 2 } 1 2 } < 0.886 42. [ 2 1.5, 1.5] by [ 2 1.5, 1.5] The curves intersect at x 5 0 and x < 6 0.9286. Use NINT to find 2 E 0.9286 0 (sin x 2 x 3 ) dx < 0.4303. 43. First graph y 5 cos x and y 5 x 2 . [ 2 1.5, 1.5] by [ 2 0.5, 1.5] The curves intersect at x < 6 0.8241. Use NINT to find 2 E 0.8241 0 (cos x 2 x 2 ) dx < 1.0948. Multiplying both functions by k will not change the x -value of any intersection point, so the area condition to be met is 2 5 2 E 0.8241 0 ( k cos x 2 kx 2 ) dx 2 5 k ?
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Unformatted text preview: 2 E 0.8241 (cos x 2 x 2 ) dx ⇒ 2 < k (1.0948) ⇒ k < 1.8269. 44. (a) Solve for y: } a x 2 2 } 1 } b y 2 2 } 5 1 y 2 5 b 2 1 1 2 } a x 2 2 } 2 y 5 6 b ! 1 § 2 § } a x § 2 2 } § (b) E a 2 a 3 b ! 1 § 2 § } a x § 2 2 } § 2 1 2 b ! 1 § 2 § } a x § 2 2 } § 24 dx or 2 E a 2 a b ! 1 § 2 § } a x § 2 2 } § dx or 4 E a b ! 1 § 2 § } a x § 2 2 } § dx (c) Answers may vary. (d, e) 2 E a 2 a b ! 1 § 2 § } a x § 2 2 } § dx 5 2 b 3 } 2 x } ! 1 § 2 § } a x § 2 2 } § 1 } a 2 } sin 2 1 } a x } 4 5 2 b 3 } a 2 } sin 2 1 (1) 2 } a 2 } sin 2 1 ( 2 1) 4 5 p ab 45. By hypothesis, f ( x ) 2 g ( x ) is the same for each region, where f ( x ) and g ( x ) represent the upper and lower edges. But then Area 5 E b a [ f ( x ) 2 g ( x )] dx will be the same for each. a 2 a 1 a 3 } } 4 3 } a 3 a Section 7.2 299...
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