sol to a9

sol to a9 - MAT 137Y 2008-09 Winter Session, Solutions to...

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Unformatted text preview: MAT 137Y 2008-09 Winter Session, Solutions to Problem Set 9 1 (SHE 6.3) 12. The volume is V = Z 1 2 π x · 2 x dx + Z 2 1 2 π x · 2 √ 2- x dx = 4 π Z 1 x 2 dx- 4 π Z 1 ( 2- u ) √ u du ( u = 2- x ) = 4 π x 3 3 1- 4 π 4 3 u 3 / 2- 2 5 u 5 / 2 1 = 76 15 π . 34. The volume is given by V = Z √ r 2- a 2 2 π x ( p r 2- x 2- a ) dx = 2 π Z √ r 2- a 2 h x ( r 2- x 2 ) 1 / 2- ax i dx = 2 π- 1 3 ( r 2- x 2 ) 3 / 2- a 2 x 2 √ r 2- a 2 = 1 3 π ( 2 r 3 + a 3- 3 ar 2 ) . 46. We let u = r 2- x 2 so du =- 2 x dx . The volume is given by V = 2 Z r q r 2- h 2 4 2 π x p r 2- x 2 dx =- 2 π Z h 2 / 4 u 1 / 2 du = 2 π 2 3 u 3 / 2 h 2 / 4 = π h 3 6 . 2 (a) The region and a typical shell is illustrated below. (b) Using shells, the volume is V = Z 1 2 π ( y + 1 )( √ y- y 2 ) dy = 2 π Z 1 y 3 / 2 + y 1 / 2- y 3- y 2 dy = 2 π 2 5 y 5 / 2 + 2 3 y 3 / 2- 1 4 y 4- 1 3 y 3 1 = 29 π 30 . (c) If we were to use washers, then the volume would be expressed as V = Z 1 π ( √ x + 1 ) 2- π ( x 2 + 1 ) 2 dx = π Z 1 x + 2 √ x + 1- x 4- 2 x 2- 1 dx = π Z 1 x + 2 √ x- x 4- 2 x 2 dx = π...
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This note was uploaded on 10/14/2010 for the course MAT MAT taught by Professor Unknown during the Spring '10 term at Touro CA.

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sol to a9 - MAT 137Y 2008-09 Winter Session, Solutions to...

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