University of Waterloo Midterm Questions     MATH 138: Winter 2012 1)For this question consider the following integral,     Decompose the integrand using Partial Fractions and determine the coefficients

University of Waterloo

Midterm Questions     MATH 138: Winter 2012

1)For this question consider the following integral,

 

 

1. 1. 1. Decompose the integrand using Partial Fractions and determine the coefficients.
      2. Use your answer from part a) to evaluate the definite integral above.

.

,

2. definite integrals

b)       

 

 

3. indefinite integrals.

a)

.

 

b)

4. indefinite integrals.

a)

b)       

 

 

5. Consider the bowl-shaped volume of revolution generated by rotating the area in the first quadrant above the curve y = x⁴ and below the line y = 2 about the y- axis.
   1. Sketch and label the curves and the area enclosed by the curves.
   2. State the integral for the volume of revolution using the method of disks/washers.
   3. State the integral for the volume of revolution using the method of shells.
   4. Evaluate ONE of the integrals in part b) or c) to find the volume of revolution.

6. Consider the definite integral,

1. 1. Evaluate this integral exactly.
   2. Use the Trapezoidal rule with n = 4 to approximate the integral. What is the error in this method?
   3. Use the Simpson’s rule with n = 4 to approximate the integral. What is the error in this method?
   4. Find a bound on the error (using the Formula sheet) for the Trapezoidal rule using the smallest value of K possible. Is your error from b) within this error?
   5. Find a bound on the error (using the Formula sheet) for the Simpson’s rule using the smallest value of K possibl Is your error from c) within this error?
   6. What characteristic of Simpson’s rule makes it more accurate than the Trapezoidal rule for this integral?

7. Evaluate the following definite integrals.

a)

b)