8.1ByPartsSolutions

8.1ByPartsSolutions - MthSc 108-Learning Activity Solutions Section 8.1 Integration by Parts 1 log 2 x dx(Hint First use a property to rewrite the

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MthSc 108-Learning Activity Solutions Section 8.1: Integration by Parts 1. 2 log xdx (Hint: First use a property to rewrite the integrand.) Solution: First note that 2 ln log ln 2 x x = . Now let ln ux = . Then 1 du dx x = . Let dv dx = and then vx = . 2 22 ln 1 1 1 log ln ln ln 2 ln 2 ln 2 1l n ln ln 2 ln 2 ln 2 log log ln 2 x x dx dx xdx x x x dx x xx xx d x x C x x x C  == =   =− = + + ∫∫ 2. Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve x ye = , and the line 1 x = about the y-axis. (Hint: Use the shell method.) Solution: 1 0 2 x Vx e d x π = Now let = . Then du dx = . Let x dv e dx = and then x ve . V = 2 xe x dx 0 1 = 2 xe x 0 1 + e x dx 0 1 = 2 1 e +− e x 0 1 = 2 1 e 1 e + 1 = 2 4 e V = 2 4 e cubic units
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3. Find the volume of the solid generated by revolving the region bounded by the x-axis and the curve sin yx x = , 0 x π ≤≤ , about the y-axis. (Hint: Use the shell
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This note was uploaded on 01/30/2011 for the course MTHSC 108 taught by Professor Any during the Spring '08 term at Clemson.

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8.1ByPartsSolutions - MthSc 108-Learning Activity Solutions Section 8.1 Integration by Parts 1 log 2 x dx(Hint First use a property to rewrite the

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