MAT 202 Final, Fall 2009

MAT 102/202 I- Calculate the following integrals: (35pts-5pts each) 1)   dx x x x 5 4 3 2)   2 2 3 4 x dx   3)  dx x x 5 ln 4)  xdx x x cos sin 5)    tan x 5dx 6)       dx x x x 3 1 3 5 7)   dx x 3 1 4 3 II- Find the following derivatives: (10pts-5pts each) 1) x x x x x y 1sec 1 2 1 1 sec 2        2)       2 3 2 3 2 1 2 4 5 tan 1 4 2           x x x x x y x III- Solve the following: (10pts-5pts each) 1) Using the shell method, find the volume of the solid generated by revolving about the y-axis, the region bounded by: x y x y   & 2 . 2) Find the length of the curve  x y  ln sec from 4 0    to x x IV_ Solve for x: (10pts-5pts each) 1)     0 6ln 5 ln ln 2 3    x x x MAT 102/202 2) 4 2 2 3 4 5     e x x e e x V- (10pts-5pts each)Test the following integrals for convergence. If they converge, find their limits: a.       0 5 1 dx x x b.      1 ln 2 ln 1 dx x x x VI- (5pts) Solve for x when 6 2 1 tan sin 2 1          x x VII- (10pts-5pts each) Evaluate the following integrals: a)  dzdydx e e e z y x    1 1 1 ln ln ln b)   1 0 1 x dydx e y x VIII- Solve the following problems: (10pts-5pts each) a. Find the acute angle between the 2 vectors: j i v j i u 2 3 4 & 2     b. Find the unit vector(s) that are parallel and normal to the vector j i v   4