Martens_WA6

# Martens_WA6 - 7.1 2 f x = x 2 2 x 1 g x = 2 x 5 âˆ âˆ 2...

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Unformatted text preview: 7.1 2. f ( x) = x 2 + 2 x + 1 g ( x) = 2 x + 5 âˆ« âˆ« 2 âˆ’2 2 ( x 2 + 2 x + 1) âˆ’ (2 x + 5) dx = x 2 âˆ’ 4 dx = 2 âˆ’2 x3 3 âˆ’ 4x âˆ’2 6. f ( x) = ( x âˆ’ 1)3 g ( x) = ( x âˆ’ 1) âˆ« âˆ« âˆ« 2 âˆ’1 2 ( x âˆ’ 1)3 âˆ’ ( x âˆ’ 1) dx = (âˆ’1 + 3 x âˆ’ x 2 + x3 ) âˆ’ ( x âˆ’ 1) dx = 2 x âˆ’ x 2 + x 3 dx = âˆ’1 2 âˆ’1 8. âˆ« ( 1 âˆ’ x ) âˆ’ ( x 1 2 âˆ’1 2 âˆ’ 1) dx (see green area) 12. âˆ«Ï€ âˆ«Ï€ âˆ’ Ï€ /4 âˆ’ /4 (sec 2 x âˆ’ cos x )dx = Ï€ /4 1 âˆ’ cos x dx /4 cos 2 x (no region represented by the integral) 20. f ( x) = âˆ’ x 2 + 4 x + 1 g ( x) = x + 1 âˆ’ x2 + 4x + 1 = x + 1 âˆ’ x2 + 4x + 1 âˆ’1 âˆ’ x = 0 âˆ’ x 2 + 3x = 0 ( x)(3 âˆ’ x ) = 0 x = 0, x = 3 âˆ« ( âˆ’ x + 4 x + 1) âˆ’ ( x + 1) dx = âˆ« âˆ’ x + 3x dx = 3 2 0 3 2 0 x3 x2 âˆ’ + 3 + C = 3 2 0 33 03 32 02 âˆ’ + 3 + C âˆ’ âˆ’ + 3 + C = 3 2 2 3 27 âˆ’9 + 2 = 13.5 âˆ’ 9 = 4.5 3 24. 1 x2 y=0 x =1 x=5 y= âˆ« 5 1 1 x 2 âˆ’ 0 dx = 5 1 ln x 2 = 1 2 1 1 ln 52 âˆ’ ln12 = 2 2 1 ln 25 â‰ˆ 2 1.60943791 28. f ( y ) = y (2 âˆ’ y ) g ( y) = âˆ’ y y (2 âˆ’ y ) = âˆ’ y 2 y âˆ’ y2 + y = 0 3y âˆ’ y2 = 0 3y âˆ’ y2 = 0 y=0 y =3 âˆ« [ ( y(2 âˆ’ y)) âˆ’ (âˆ’ y)]dy = âˆ« 2 y âˆ’ y + y dy = âˆ« 3 y âˆ’ y dy = 3 0 3 2 0 3 2 0 3 y 2 y3 2 âˆ’ 3 = 0 3(3) 2 (3)3 3(0) 2 (0)3 âˆ’ âˆ’ âˆ’ = 3 2 3 2 27 27 âˆ’ = 2 3 13.5 âˆ’ 9 = 4.5 3 30. f ( y) = g ( y) = 0 y =3 y 16 âˆ’ y 2 y âˆ«0 16 âˆ’ y 2 dy = 3 y âˆ«0 4 âˆ’ y dy = 3 âˆ« 3 y âˆ’ y dy + âˆ«0 [ 4] dy = 0 3 3 3 [ x] 0 + [ 4x] 0 = (3 âˆ’ 0) + (4(3) âˆ’ 0) = (3) + (12) = 15 34. f ( x) = x 3 âˆ’ 2 x + 1 g ( x ) = âˆ’2 x x =1 âˆ« ( x âˆ’ 2 x + 1) âˆ’ ( âˆ’2 x ) dx = âˆ« x + 1 dx = 1 3 âˆ’1 1 3 âˆ’1 x4 4 + x = âˆ’1 14 âˆ’14 âˆ’ 1 = + 1 âˆ’ 4 4 5 3 âˆ’âˆ’ = 4 4 2 1 36. y = x4 âˆ’ 2x2 y = 2x2 x=0 x=2 x = âˆ’2 âˆ« ( x âˆ’ 2 x ) âˆ’ ( 2 x ) dx âˆ’ âˆ« ( 2 x ) âˆ’ ( x âˆ« x âˆ’ 4 x dx âˆ’ âˆ« 4 x âˆ’ x dx 0 4 2 2 2 2 âˆ’2 0 0 4 2 2 2 4 âˆ’2 0 4 âˆ’ 2 x 2 ) dx x5 4 x3 4 x3 x5 âˆ’ âˆ’ âˆ’ 5 3 âˆ’2 3 5 0 0 2 (0)5 4(0)3 (âˆ’2)5 4(âˆ’2)3 4(2)3 (2)5 4(0)3 (0)5 âˆ’ âˆ’ âˆ’ âˆ’ âˆ’ âˆ’ âˆ’ 35 3 3 5 3 5 5 64 64 + 15 15 40. f ( x) = y=0 0â‰¤ xâ‰¤3 6x x +1 2 6x x 2 + 1 dx = 3 6x 3âˆ« 2 dx 0 3x + 1 âˆ« 3 0 ln(3 x 2 + 1) 0 ln(3(3) 2 + 1) âˆ’ ln(3(0) 2 + 1) ln(28) âˆ’ ln(1) ln(28) â‰ˆ 3.33220451 3 50. f ( x) = 2sin x + cos 2 x âˆ« ( 2sin x + cos 2 x ) dx 0 Ï€ 1 âˆ’2 cos x + 2 sin(2 x) 0 Ï€ 1 1 âˆ’2 cos Ï€ + sin(2Ï€ ) âˆ’ âˆ’2 cos 0 + sin(2(0)) 2 2 1 âˆ’2 cos Ï€ + sin(2Ï€ ) + 2 2 2+0+2 4 ...
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## This note was uploaded on 12/30/2009 for the course MAT 231 taught by Professor Thurber during the Spring '09 term at Thomas Edison State.

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