Calculus 1  Hour Exam 3   -   Math 221     Fall, 2011

 

 

1.

 

 

 

Evaluate the integrals:    a. ∫ (x -2  + x4)dx   b.  ∫ (x sin( x2) dx    c. ∫ (x +1)(x – 2)dx   

 

Answer:  a. -1/x + (1/5) x5 + C       b. – ½ cos (x2) + C       c.  (1/3)x3 – ½ x2 – 2x  + C

 

 

 

 

2.

 

                                                         0                               1

Evaluate each definite integral:  a.  ∫ (9x2  - 4 x3)dx     b.  ∫ [ (1 / (x2 + 1)]dx  

                                                       -1                                0

         2

c.      ∫ ( e1/x ( - 1/x2)dx  

        1

 

Answers:      a.   4                        b.  π/4                      c.   √e  –  e

 

 

 

3.

 

 

Find the area of the region enclosed by  y = ( x – 2)2  and  y = x.

 

Answers:  9/2

 

 

 

4.

 

 

a.                   State the Fundamental Theorem of Calculus, Part 1.

 

b.                  State the Fundamental Theorem of Calculus, Part 2.

 

 

 

 

 

5.

 

Set up, but do not evaluate, an integral for the volume of the solid whose base is the

triangular region with vertices (0,0), (1,0), and (0,1) and whose cross-sections

perpendicular to the x-axis are squares.

 

                1

Answer:  ∫ ( -x + 1)2 dx

               0

 

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