Calculus 2  Hour Exam 2   -   Math 231     Spring, 2012

 

 

 

 

 

 

 

 

 

 

 

5.

 

 

Consider the curve  y  =  4x2 + 3  between the points (0,3) and (1,7).  Set up

But do not evaluate integrals which represent the following quantities.

 

a.       The length of the curve.

 

b.      The surface area when the curve is rotated about the x-axis.  Your

Answer must be an integral with respect to  x.

 

c.       The same as part  b), but this time, your answer must be an integral with

respect to y.

                   1                                            1

Answers:    L  =  ∫ √ [ 1 + (8x)2 ] dx         SA =  ∫ 2π (4x2 + 3) √[ 1 + 64x2 ] dx

                              0                                            0

 

                             7

                  SA =  ∫ 2π y √ [ 1 + (1/16) {1 / (y – 3) } ] dy

                            3

 

 

 

 

 

 

6.

 

Find a value of  N  which guarantees that the partial sum   SN  approximates

The series                   ∞

                                    Σ 1/n5

                                  n=1

 

to within  1/1000.  Give brief but complete details. 

 

(Hint:  use the ideas about the integral test.)      Answer:  N  ≥ 2501/4

 

 

 

 

 

7.

 

 

 

Use the comparison test or limit comparison test to determine if the series converge or

diverge.  Show all details of your argument.  State precisely which series you are

comparing with the original series.

 

            ∞                                                            ∞

a.       Σ (2n + n sin n) / (n2 + 10n + 11)      b.  Σ e-n / n

n = 1                                                     n=1

 

Answers:     a. Diverges      b.  Converges