UMUC MATH, SPRING 2017 FINAL KOBI SNITZ (1) Consider a sphere of radius 3, centered at the origin. The top of the sphere is cut off at height of y = 2 and then a cylindrical hole with radius of 1 centered on the y-axis is drilled from flatten top of the sphere to a depth of 1. Calculate the area of a slice of the sphere at height y. Use your formula for the area of the slice to calculate the volume of the remaining part of the sphere with the hole in it. (2) let (t) = (2t3 1, 3t2 +4) be a parametrized curve. Calculate the arclength of the curve from the pointRt = 0 to the point t = 4. (3) Find the anti derivative of 5+x12 +2x R (4) For any n write the anti derivative of x cosn (x) dx in terms of an integral involving cosn1 (x). That is, find a reduction formula for this kind of integral. (5) for which x does the series converge 1 1 1 + + + ... (x 3) (x 3)3 (x 3)5 when the series does converge, what is the value it converges to (6) write a power series representation of x3 sin(x). Then use the the series to write a power series for the integral Z 1 x3 sin(x) 0 1