Problems appearing on your in-class final will be similar to those here but will have numbers and functions changed.

Here?s an example of the way problems selected may be changed for your in-class final exam. If problems 1 and 3 were selected, they might appear like this:

#1 modified
A water tank has the shape of the surface generated by revolving the parabola segment , about the y-axis.  If the tank is filled to a depth of 6 feet with a fluid weighing 80 lbs per cubic foot, find the work W required to pump the contents of the tank to a height of 4 feet above the top of the tank.

#2 modified
. . . below by the curve y = cos x, 0 ≤ x ≤ π/2, . . .with the remainder of the question as printed.

1.   [Answer]	A water tank has the shape of the surface generated by revolving the parabola segment about the y-axis. If the tank is filled to a depth of 8 feet with a fluid weighing 80 lbs. per cubic foot, find the work W required to pump the contents of the tank to a height of 3 feet above the top of the tank.
2.   [Answer]	Find the exact value of the volume generated by rotating the region in the first quadrant bounded above by the line , on the left by the y-axis, and below by the curve y = tan x, 0 ≤ x ≤ π/3, about (a) the x-axis; (b) the line
3.   [Answer]	The design of a new airplane requires a gasoline tank of constant cross-section area in each wing. The tank must hold 5000 lb. of gasoline that weighs 42 lb/ft3.
a.	A scale drawing of a cross section of the wing is shown. Estimate the length of the tank.   y0 = 2.1 ft., y1 = 2.1 ft., y2 = 2.0 ft, y3 = 1.9 ft, y4 = 1.8 ft, y5 = 1.6 ft, y6 = 1.5 ft.; Horizontal spacing = 1 ft. Use (for n subintervals of length )
b.	A cross section of a wing is in the shape of the area below on the interval [0,7.5]. The presence of wires and other mechanisms mean the tank must fit in the interval from [0.3, 6.3]. Use six trapezoids to estimate the area and calculate the length of the tank.
c.	If the length of the tank (and the wing) is to be accurate to the nearest tenth of an inch (and all other values are assumed to be exact), how many trapezoids would be needed to estimate the area of the tank cross section? Hint: To have the desired accuracy, the area of a cross section must have 3 significant figures, or accuracy to the nearest hundredth.
4.   [Answer]	Find the exact value of the length of the curve x = e2t - , y = et for 0 ≤ t ≤ ln 2. Completely simplify your answer.
5.   [Answer]	Evaluate the following integrals. Show the details of substitutions you make in reaching your answer.
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
6.   [Answer]	Sketch the curve y = Arc cos . Find the area between this curve and the x-axis from x = -2 to x = 2.
7.   [Answer]	Compute the volume V and the surface area S of the solid formed by rotating the area between the curve and the x-axis (for 1 ≤ x < ∞) around the x-axis. What seemingly paradoxical property does this solid possess? Sketch.
8.   [Answer]	Interpret the integrals as areas and use the result to express the sum above as one definite integral. Evaluate the new integral.
9.   [Answer]	Suppose {bn} is a sequence of positive numbers converging to m ≠ 0, with b1 = 1. Prove that the series converges to a sum expressible in terms of m.
10.   [Answer]	Find the Taylor series expansion of f(x) = cos x about π/2. Write the series in sigma notation in two forms: the first with index value starting at zero, and the second with index value starting at one.
11.   [Answer]	Determine whether the following series are absolutely convergent, convergent, or divergent. Specify which tests you use and show all relevant reasoning.
a.
b.
c.
d.
e.
f.
12.   [Answer]	The base of a certain solid is the region of the xy-plane bounded by y = x2 and y = x. Every cross section perpendicular to the x-axis is a semicircle with a diameter in the base. Find the exact volume of this solid.
13.   [Answer]	Determine whether converges. If it converges, find its exact sum.
14.   [Answer]	Use the minimum number of series terms needed to compute with an error of magnitude less than 10-7. Show how you know when you?ve used the minimum number of terms.
15.   [Answer]	Find the center of mass coordinates for the thin plate of constant density covering the region bounded by y = ln x, y = -4, x = 1, and x = 2.
16.   [Answer]	Find the interval of convergence, clearly showing testing details and names of tests used at interval endpoints.
a.
b.
17.   [Answer]	Find exactly the volume generated when the area bounded by , the x-axis, y-axis, and x = 1 is revolved about the y-axis. Simplify your answer.
18.   [Answer]	Write the equation of the plane containing the points (0,2,3) and (1,-3,7) and orthogonal to the plane 2x + y − z = 5..
19.   [Answer]	Find the distance between the lines and .
20.   [Answer]	Find the parametric equations of the line through the point (3,6,4) that intersects the z-axis and is parallel to the plane x − 3y + 5z = 6 .
21.   [Answer]	The force F (in Newtons) of a hydraulic cylinder in a press is proportional to the square of sec x , where x is the distance (in meters) that the cylinder is extended in its cycle. The domain of F is [0,π/3] and F(0) = 500.
a.   [Answer]	Find F as a function of x.
b.   [Answer]	Find the average force exerted by the press over the interval [0 ,π/3].
22.   [Answer]	A weight of 255 pounds is to be lifted straight up with ropes at angles of 20 and 30 degrees to the direction of travel as shown below. Find the tension in each rope.
23.   [Answer]	Find the point in which the line through the origin perpendicular to the plane 2x − y − z = 4 intersects the plane 3x − 5y + 2z = 6 .
24.   [Answer]	Find all numbers b and c such that the vectors 2bi -2j + k and ci + 2j + 2k are orthogonal and have the same magnitude.
25.   [Answer]	Express the vector a = 2i + 4j + 5k as the sum of a vector p which is parallel to the vector b = 2i − j − 2k and a vector n which is normal to p.
26.   [Answer]	Find the volume of the solid formed by rotating the region bounded by , the x-axis, and the lines x = ±3 about the x-axis.
27.   [Answer]	Use the Maclaurin series representation of to find a Maclaurin representation of f(x) = xln(1 + x) which converges for . Write the series in sigma notation in two forms: the first with index value starting at zero, and the second with index value starting at two.
28.   [Answer]	a.    Find the Maclaurin series for .   b.    Use the series to estimate the value of with error of magnitude less than 0.00005.

1.   back 74240 πft-lbs 2.   back a. cubic units              b. 3.   back a. ≈ 10.63 feet              b. ≈ 14.24 feet ;              c. n ≥ 66 4.   back units 5.   back a.              b.              c. 56              d.              e.              f.              g.              h.              i. converges to 2              j. diverges 6.   back 2π 7.   back solid has a finite volume of π, but an infinite surface area 8.   back 9.   back 10.   back

11.   back a. converges absolutely because              b. diverges              c. converges absolutely              d. converges              e. converges absolutely              f. converges absolutely 12.   back cubic units 13.   back converges to ½ 14.   back 0.1995560889 15.   back (1.513, -1.802) 16.   back a. [2,4)              b. 17.   back cubic units 18.   back x + 9y + 11z = 51 19.   back 20.   back x = 3t + 3; y = 6t + 6; z = 3t + 4 21.   back a. F(x) = 500sec2x              b. 827 newtons 22.   back rope at 30°: 114 lb              rope at 20°: 166 lb 23.   back (4/3, -2/3,-2/3) 24.   back b = c = 1; b = c = -1 25.   back              26.   back 27.   back 28.   back              b. 0.5351