MATHEMATICS H161--AU03    MIDTERM ONE

 

NAME _____________________________

 

 

Do all problems.  Show your work.  (Even if you get an answer just from a calculator, you need to explain why it is correct.)

 

1.  Calculate the following definite integrals.

 

            (a) (5 points) The integral from  0  to  pi/2 of the integrand  cos3 x.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

            (b) (5 points) The integral from  0  to  1  of the integrand  (x3 multiplied by the square root of  1 x2).

 

 

 

 

 

 

           

                                                     

 

 

 


2. Find the following indefinite integrals.

 

(a) (5 points) The integrand is  (2x 1)/square root of (4x2 4x + 3)

 

 

 

 

 

 

 

 

 

 

 

(b) (5 points) The integrand is  x/square root of (x - 2)

 

 

 

 

 

 

 

 

 

 

 

 

3.  (10 points)  Find the area of the region bounded by the straight line  y = x + 2  and the parabola  y = 4 x2  from  x = 2  to  x = 1.

 

 

 

 

 

 

                                               


4.  (a) (10 points)  The region bounded by the parabola  y = 2x2    and the straight line  y = 2  is rotated about the y-axis.  Find the volume of the resulting solid.

 

 

 

 

 

 

 

 

 

 

 

 

 

(b)  (10 points) The same region as in  (a)  is rotated about the  x-axis.  Find the volume of the resulting solid.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5.  (10 points)  Let  f(x) = 2x + 1.  Prove using an epsilon/delta argument that  limit as x approaches 1  of   f(x) is  1.