Calculus 1  Hour Exam 3  (continued)  -   Math 220     Fall, 2010

 

 

 

 

 

 

7.

 

 

                                                                                                        12

Suppose that  f  is integrable on the interval  [2, 12].  Given that  ∫ f(x) dx  =  25,

                                                                                                        2

8                                 12

∫ f(x) dx  =  10  and    ∫ f(x) dx  =  22,  evaluate the following definite integrals.

2                                  4

 

            2                       4                         8

a.       ∫ f(x) dx       b.  ∫ f(x) dx         c.  ∫ f(x) dx      

8                       2                         4

 

                                                                               Answers:  a.  -10         b.  3         c.  7                            

 

 

 

 

 

 

 

 

 

8.

 

Let  R  be the region bounded above by graph of  y  =  sin x / x  and bounded below by

the x-axis on the interval [ 2π, 3π ].  Set up, but do not evaluate, definite integrals which

represent the given quantities.  User proper notation.

 

a.  The area of   R .

 

b. The volume of the solid obtained when  R  is revolved around the x-axis.

 

c. The volume of the solid obtained when  R  is revolved around the vertical line  x = 3.

 

                         3π                                  3π

Answers:      a.  ∫  sin x / x  dx           b.  ∫ π ( sin x / x)2  dx

                        2π                                  2π

 

            3π

c.       ∫  [ 2π (x – 3) ( sin x / x ) ]  dx

           2π

 

 

 

9.

 

 

Suppose  F(x)  is a polynomial with  F ‘ (x)  =  f(x) .  Given that  F(0)  =  2,  F(2)  = 8,

F(4)  =  28,  F(6)  =  68, and  F(8)  = 42.  Fomd the average value of  f(x)  on the

interval  [ 2, 6 ].

                                                                                Answer:    15