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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

Page1 / 4

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online