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

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Unformatted text preview: 7.1 2. f ( x) = x 2 + 2 x + 1 g ( x) = 2 x + 5 ∫ ∫ 2 −2 2 ( x 2 + 2 x + 1) − (2 x + 5) dx = x 2 − 4 dx = 2 −2 x3 3 − 4x −2 6. f ( x) = ( x − 1)3 g ( x) = ( x − 1) ∫ ∫ ∫ 2 −1 2 ( x − 1)3 − ( x − 1) dx = (−1 + 3 x − x 2 + x3 ) − ( x − 1) dx = 2 x − x 2 + x 3 dx = −1 2 −1 8. ∫ ( 1 − x ) − ( x 1 2 −1 2 − 1) dx (see green area) 12. ∫π ∫π − π /4 − /4 (sec 2 x − cos x )dx = π /4 1 − cos x dx /4 cos 2 x (no region represented by the integral) 20. f ( x) = − x 2 + 4 x + 1 g ( x) = x + 1 − x2 + 4x + 1 = x + 1 − x2 + 4x + 1 −1 − x = 0 − x 2 + 3x = 0 ( x)(3 − x ) = 0 x = 0, x = 3 ∫ ( − x + 4 x + 1) − ( x + 1) dx = ∫ − x + 3x dx = 3 2 0 3 2 0 x3 x2 − + 3 + C = 3 2 0 33 03 32 02 − + 3 + C − − + 3 + C = 3 2 2 3 27 −9 + 2 = 13.5 − 9 = 4.5 3 24. 1 x2 y=0 x =1 x=5 y= ∫ 5 1 1 x 2 − 0 dx = 5 1 ln x 2 = 1 2 1 1 ln 52 − ln12 = 2 2 1 ln 25 ≈ 2 1.60943791 28. f ( y ) = y (2 − y ) g ( y) = − y y (2 − y ) = − y 2 y − y2 + y = 0 3y − y2 = 0 3y − y2 = 0 y=0 y =3 ∫ [ ( y(2 − y)) − (− y)]dy = ∫ 2 y − y + y dy = ∫ 3 y − y dy = 3 0 3 2 0 3 2 0 3 y 2 y3 2 − 3 = 0 3(3) 2 (3)3 3(0) 2 (0)3 − − − = 3 2 3 2 27 27 − = 2 3 13.5 − 9 = 4.5 3 30. f ( y) = g ( y) = 0 y =3 y 16 − y 2 y ∫0 16 − y 2 dy = 3 y ∫0 4 − y dy = 3 ∫ 3 y − y dy + ∫0 [ 4] dy = 0 3 3 3 [ x] 0 + [ 4x] 0 = (3 − 0) + (4(3) − 0) = (3) + (12) = 15 34. f ( x) = x 3 − 2 x + 1 g ( x ) = −2 x x =1 ∫ ( x − 2 x + 1) − ( −2 x ) dx = ∫ x + 1 dx = 1 3 −1 1 3 −1 x4 4 + x = −1 14 −14 − 1 = + 1 − 4 4 5 3 −− = 4 4 2 1 36. y = x4 − 2x2 y = 2x2 x=0 x=2 x = −2 ∫ ( x − 2 x ) − ( 2 x ) dx − ∫ ( 2 x ) − ( x ∫ x − 4 x dx − ∫ 4 x − x dx 0 4 2 2 2 2 −2 0 0 4 2 2 2 4 −2 0 4 − 2 x 2 ) dx x5 4 x3 4 x3 x5 − − − 5 3 −2 3 5 0 0 2 (0)5 4(0)3 (−2)5 4(−2)3 4(2)3 (2)5 4(0)3 (0)5 − − − − − − − 35 3 3 5 3 5 5 64 64 + 15 15 40. f ( x) = y=0 0≤ x≤3 6x x +1 2 6x x 2 + 1 dx = 3 6x 3∫ 2 dx 0 3x + 1 ∫ 3 0 ln(3 x 2 + 1) 0 ln(3(3) 2 + 1) − ln(3(0) 2 + 1) ln(28) − ln(1) ln(28) ≈ 3.33220451 3 50. f ( x) = 2sin x + cos 2 x ∫ ( 2sin x + cos 2 x ) dx 0 π 1 −2 cos x + 2 sin(2 x) 0 π 1 1 −2 cos π + sin(2π ) − −2 cos 0 + sin(2(0)) 2 2 1 −2 cos π + sin(2π ) + 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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