# 3 - MATH 138 Winter 2011 Assignment 3 Topics Volumes by...

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Unformatted text preview: MATH 138 Winter 2011 Assignment 3 Topics: Volumes by disks, volumes by cylindrical shells. Due: 11 a.m. Friday, January 28. Hand in your solutions to the following problems. 1. Evaluate the following integrals: (a) R 1 x 3- 4 x- 10 x 2- x- 6 dx x 3- 4 x- 10 x 2- x- 6 = x + 1 + 3 x- 4 ( x- 3)(3+2) . We write 3 x- 4 ( x- 3)(3+2) = A x- 3 + B x +2 . Then 3 x- 4 = A ( x + 2) + B ( x- 3) = ( A + B ) x + 2 A- 3 B ⇒ A + B = 3 and 2 A- 3 B =- 4 ⇒ A = 1 and B = 2 . Thus R 1 x 3- 4 x- 10 x 2- x- 6 dx = R 1 ( x + 1 + 1 x- 3 + 2 x +2 ) dx = ( 1 2 x 2 + x + ln | x- 3 | + 2 ln | x + 2 | ) 1 = ( 1 2 +1+ ln 2+2 ln 3)- (0+0+ ln 3+2 ln 2) = 3 2 + ln 3- ln 2 = 3 2 + ln 3 2 . (b) R x 2- 2 x- 1 ( x- 1) 2 ( x 2 +1) dx We write x 2- 2 x- 1 ( x- 1) 2 ( x 2 +1) = A ( x- 1) + B ( x- 1) 2 + Cx + D ( x 2 +1) ⇒ x 2- 2 x- 1 = A ( x- 1)( x 2 + 1) + B ( x 2 + 1) + ( Cx + D )( x- 1) 2 = x 3 ( A + C ) + x 2 (- A + B- 2 C + D ) + x ( A + C- 2 D ) + ( A + B + D ) ⇒ A = 1 ,B =- 1 ,C =- 1 ,D = 1 . Thus R x 2- 2 x- 1 ( x- 1) 2 ( x 2 +1) dx = R 1 ( x- 1)- 1 ( x- 1) 2- x ( x 2 +1) + 1 ( x 2 +1) dx = ln | x- 1 | + 1 x- 1- 1 2 ln ( x 2 + 1) + tan- 1 x + C 2. Make a substitution to express the integrand of R cos x sin 2 x + sin x dx as a rational function and then evaluate the integral. Let u = sin x . Then du = cos xdx so R cos x sin 2 x + sin x dx = R du u 2 + u = R 1 u du- R 1 u +1 du = ln | u u +1 | + C = ln | sin x ( sin x )+1 | + C ....
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3 - MATH 138 Winter 2011 Assignment 3 Topics Volumes by...

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