In problems 1 – 6, use the values given in the tables to calculate the volumes of the solids. (Fig. 28 – 33)
In problems 7 – 12, represent each volume as an integral and evaluate the integral.
9. Fig. 36. For 1 ≤ x ≤ 4, each face is a triangle with base x + 1 meters and height x meters.
The graphs of the functions in problems 5 – 8 are straight lines. Calculate each length (a) using the distance formula between 2 points and (b) by setting up and evaluating the arc length integrals.
5) y = 1 + 2x for 0 ≤ x ≤ 2
7) x = 2 + t , y = 1 – 2t for 0 ≤ t ≤ 3
In problems 15 – 23, (a) represent each length as a definite integral, and (b) evaluate the integral using your calculator's integral command
15. The length of y = x2 from (0,0) to (1,1)
17. The length of y = x from (1,1) to (9,3)
1. A tank 4 feet long, 3 feet wide and 7 feet tall (Fig. 15) is filled with water which weighs 62.5 pounds per cubic foot. How much work is done pumping the water out over the top of the tank?
3. A tank 5 feet long, 2 feet wide and 4 feet tall is filled with a oil which weighs 60 pounds per cubic foot.
(a) How much work is done pumping all of the oil out over the top edge of the tank?
(b) How much work is done pumping the top 36 cubic feet of oil out over the top edge of the tank?
(c) How long does a 1 horsepower pump take to empty the tank over the top edge of the tank? (A 1 horsepower pump works at a rate of 33,000 foot–pounds per minute.) A 1/2 horsepower pump? Which pump does more work?
9. If you and a friend share the work equally in emptying the conical container in Problem 8, what depth of grain should the first person leave for the second person to empty?
In problems 11 – 26, sketch the region bounded between the given functions on the interval and calculate the centroid of each region (use Simpson's rule with n = 20 if necessary). Plot the location of the centroid on your sketch of the region.
11. . y = x and the x–axis for 0 ≤ x ≤ 3.
13. y = x 2 and the line y = 4 for –2 ≤ x ≤ 2
15. y = 4 – x2 and the x–axis for –2 ≤ x ≤ 2
17. y = 9 – x and y = 3 for 0 ≤ x ≤ 3
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