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10Bquiz4sol

# 10Bquiz4sol - x-axis(b Find the volume of the solid of...

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Quiz 4 Solution for MATH 10 B, 2010 1 . Calculate the integral, if it converges. Z 2 1 1 x ln x dx Solution. Let u = ln x , du = 1 x du . Then Z 2 1 1 x ln x dx = Z ln 2 ln 1 1 u du = Z ln 2 0 1 u du = lim a 0 + Z ln 2 a 1 u du = lim a 0 + [ ln | u | ] ln 2 a = lim a 0 + ( ln ( ln 2 ) - ln a ) = ( since lim a 0 + ln a = - ) Thus, it diverges.

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2 . The region bounded by y = x 2 , y = x , x = 0 and x = 1 is rotated around the x -axis. (a) Sketch the solid obtained by rotating the region around the
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Unformatted text preview: x-axis. (b) Find the volume of the solid of revolution. Solution. Volume = lim 4 x → ± ∑ π ( x 2-x 4 ) 4 x ² = π Z 1 ( x 2-x 4 ) dx = π ³ 1 3 x 3-1 5 x 5 ´ 1 = π µ 1 3-1 5 ¶ = 2 15 π...
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