Sample Hour Exam

Math 31B

The hour exam will have 8 questions. They will cover the homework up to and including October 15.

1. The graph below is the graph of a function f(x) on the domain [1,3].

The Riemann sums for the left endpoint, right endpoint, midpoint, and trapezoidal sums are, in some order: 6.208852, 6.603778, 6.600176 , 6.998705. (All are for the same N)

(a) Identify each sum which each number.

(b) Does the trapezoidal sum overestimate or underestimate the integral
? Why?

solution

2. Estimate by Simpsons Rule with N = 4, using the tabulated values

x |
0 |
0.25 |
0.5 |
0.75 |
1 |

1 |
1.22 |
1.31 |
1.37 |
1.41 |

Carry out your computations to two decimal places.

solution3. Find the area bounded by the two curves
and
y^{2} = x+4.

4. Sketch the region R bounded by the x-axis, and the line x = 1, then find the volume of the solid generated by rotating R around the line x =2.

5. Find the volume generated by rotating the region bounded curve

y = (1-x)(3-x), 1 <= x <= 3 and x-axis from 1 to 3

(a) around the y-axis. (b) around the line x = 2

solution6. A tank, formed by rotating the curve around the y-axis, is filled with water, which weighs 62.5 pounds per cubic foot. How much work is done by pumping all the water in the tank to a point 10 foot above the top of the tank?

solution7. What is the average value of the function f(x) = sin(2 x) on the interval ?

solution8. Prove, by rotating the appropriate curve, that the volume of a sphere of radius r is

solution