F 2 ln y y c 8 evaluate the improper integral

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(f) 2 ln | y | - y + C 8. Evaluate the improper integral integraldisplay 0 x 2 ( x 3 + 1) 3 dx . (a) 1 6 (b) π 2 (c) π 4 (d) π (e) 2 2 27 (f) It is divergent. 9. The curve defined by y = 1 + 2 x , between x = 1 and x = 3, is rotated about x -axis. Find the resulting surface area. parenleftbigg 1 3 parenrightbigg 1
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10. Consider the region in the first quadrant that lies below the graph of y = sin( x 2 ) + cos( x 2 ), to the left of x = radicalbig π/ 2. Let A be the area of this region. Then the x -coordinate of the center of mass of this region is 11. Suppose that a certain probability density function f ( x ) is given by: f ( x ) = Ax (1 + x 2 ) 2 for x 0; f ( x ) = 0 for x 0 , for some constant A . Then the probability that the corresponding random variable lies between - 1 and 1 is:
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