Calculus 1  Hour Exam 3   -   Math 221     Fall, 2011

 

 

 

 

 

 

6.

 

 

 

Use the shell method to set up, but do not evaluate, an integral for the volume

of the solid by rotating, about the y-axis, the region enclosed by  y = (x -1)(x-3)2

and  y = 0.  The region is shown below for your convenience.

                         

                                     3

Answer:  Volume  =     ∫ 2π x (x-1)(x-3)2 dx

                                    1

 

 

 

 

 

7.

 

                                      b                        n

Use the definition of    ∫ f(x) dx   =  lim Σ f(xi) Δx  to show that

                                     a                n→∞

          3

          ∫ ( 4 – 2x)dx  =  3

         0

 

 

 

 

 

 

 

 

 

 

 

8.

 

 

Answer true or false for each part.  No explanation is required and no partial credit.

 

a.             If the function  f  is continuous everywhere, then

 

                                                 x2

                                     d/dx [  ∫ f(t) dt ]  =  2x f(x2)    

                                                a

 

b.              If the function f  is continuous everywhere, then f has an antiderivative.

 

                              b                  

c.                       If   [  ∫ f(t) dt  ≤ 0,  then  f(x)  ≤  0  for all  x  in  [a,b]

                             a

                                                                                      b                          b

d.      If the function  f  is continuous on [a,b], then    ∫ x f(x) dx    =    x ∫ f(x) dx

                                                                                     a                          a

 

Answers:      a. True          b.  True             c.  False               d.  False