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,

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. | |

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, x ≤ π/3, | |

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/ft^{3}. | |

A scale drawing of a cross section of the wing is shown. Estimate the length of the tank. _{0} = 2.1 ft.,_{1} = 2.1 ft.,_{2} = 2.0 ft,_{3} = 1.9 ft,_{4} = 1.8 ft,_{5} = 1.6 ft,_{6} = 1.5 ft.;Use (for n subintervals of length ) | |

A cross section of a wing is in the shape of the area below on the interval | |

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. | |

Find the exact value of the length of the curve x = e^{2t} - , ^{t} | |

Evaluate the following integrals. Show the details of substitutions you make in reaching your answer. | |

Sketch the curve y = Arc cos . Find the area between this curve and the x-axis from x = -2 to x = 2. | |

Compute the volume V and the surface area S of the solid formed by rotating the area between the curve and the x-axis | |

Interpret the integrals as areas and use the result to express the sum above as one definite integral. Evaluate the new integral. | |

Suppose {b_{n}} is a sequence of positive numbers converging to m ≠ 0, with _{1} = 1.converges to a sum expressible in terms of m. | |

Find the Taylor series expansion of f(x) = cos x | |

Determine whether the following series are absolutely convergent, convergent, or divergent. Specify which tests you use and show all relevant reasoning. | |

The base of a certain solid is the region of the xy-plane bounded by ^{2} | |

Determine whether converges. If it converges, find its exact sum. | |

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. | |

Find the center of mass coordinates for the thin plate of constant density covering the region bounded by | |

Find the interval of convergence, clearly showing testing details and names of tests used at interval endpoints. | |

Find exactly the volume generated when the area bounded by , the | |

Write the equation of the plane containing the points x + y − z = 5. | |

Find the distance between the lines and . | |

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 . | |

The force F (in Newtons) of a hydraulic cylinder in a press is proportional to the square of x ,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. | |

Find F as a function of x. | |

Find the average force exerted by the press over the interval [0 ,π/3]. | |

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. | |

Find the point in which the line through the origin perpendicular to the plane x − y − z = 4x − y + 2z = 6 | |

Find all numbers b and c such that the vectors bi -2j + kci + 2j + 2k | |

Express the vector a = 2i + 4j + 5kp which is parallel to the vector b = 2i − j − 2kn which is normal to p. | |

Find the volume of the solid formed by rotating the region bounded by , the x-axis,x = ±3x-axis. | |

Use the Maclaurin series representation of to find a Maclaurin representation of f(x) = xln(1 + x) | |

a. Find the Maclaurin series for . b. Use the series to estimate the value of with error of magnitude less than 0.00005. |

b. b. ≈ 14.24 feet ; c. n ≥ 66
b. c. 56 d. e. f. g. h. i. converges to 2 j. diverges |
b. diverges c. converges absolutely d. converges e. converges absolutely f. converges absolutely b. x + 9y + 11z = 51x = 3t + 3; y = 6t + 6; z = 3t + 4F(x) = 500sec^{2}xb. 827 newtons b = c = 1; b = c = -1b. 0.5351 |