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8.1ByPartsSolutions

8.1ByPartsSolutions - MthSc 108-Learning Activity Solutions...

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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 ln2 x x = . Now let ln u x = . Then 1 du dx x = . Let dv dx = and then v x = . 2 2 2 ln 1 1 1 log ln ln ln2 ln2 ln2 1 ln ln ln2 ln2 ln2 log log ln2 x xdx dx xdx x x x dx x x x x x dx x C x xdx 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 y e = , and the line 1 x = about the y-axis. (Hint: Use the shell method.) Solution: 1 0 2 x V xe dx π = Now let u x = . Then du dx = . Let x dv e dx = and then x v e = − . V = 2 π xe x dx 0 1 = 2 π xe x 0 1 + e x dx 0 1 = 2 π 1 e + e x 0 1 = 2 π 1 e 1 e + 1 = 2 π 4 π e V = 2 π 4 π 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 y x x = , 0 x
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