School of Mathematics and Statistics
Carleton University
Math. 69.107
SOLUTIONS TO OLD TEST 3

Print Name:

Student Number: This test is out of a Total of 30.

PART I: Multiple Choice Questions
(Choose and CIRCLE only ONE answer)

1. [2 marks] Which of the following expressions gives the area of the region bounded by the curves y = -x , and between the lines x=1 and x=4?

   (a)
   

   (b)
   

   (c)
   

   
   

2. [2 marks] Evaluate the improper integral .

   (a)

   (b)

   (c)

   (d) .

3. [2 marks] Which of the following expressions gives the volume of the solid of revolution obtained when the region bounded by the graphs of y = 2x and is revolved about the y-axis?

   (a)
   

   
   

   (c)
   

   (d)
   

4. [2 marks] Evaluate

   (a)

   (b)

   

   (d)

   Use a trig. identity and a substitution

5. [2 marks] Answer TRUE or FALSE:

   The value of the improper integral is given by I = 1

   , (b) FALSE

PART II: Show all work here.
No additional pages will be accepted

1. [5+5 marks] Evaluate the following integrals using any method:

   a) .

   Solution: Let . Then . When x=0, u=0 and when , u=0. and so

   

   Alternately,

   

   Thus,

   

   b) Evaluate

   Use the Table Method:

   

   and the final answer can be written as,

   

2. Evaluate the following integrals using any method.

   [4 marks] (a)

   Solution: Write

   

   Integrate by Parts twice or use the "MYCAR" trick! You'll get

   

   where the entry in the Box is given by . See Section 8.3.4 for details. This I is an antiderivative, and so

   

   [4 marks] (b)

   Solution: The degree of the numerator exceeds that of the denominator and so we must divide these expressions. Thus

   

   Next, we use partial fractions on the third integral thus:

   

   where we let , , in the second integral (and didn't have to use partial fractions).

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