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?
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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.

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3. Find the area bounded by the two curves and y2 = x+4.

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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.

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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

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6. 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?

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

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

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