Download Workshop #3, Math 162 (Spring 2017) Week of February 6th, 2017

Workshop #3, Math 162 (Spring 2017)
Week of February 6th, 2017.
Today’s topics:
• Work
• Average value of a function
Warm-up question
Set up an integral to find the volume of the solid generated by rotating the region bounded by y = x3 − x and y = 0 for
0 ≤ x ≤ 1 about the x-axis using any method you want. Then do the same for rotating the region about the y-axis.
Workshop questions
1. A 20 ft cable weighs 100 lbs and hangs from the ceiling of a building without touching the floor. Determine the
work that must be done to lift the bottom end of the cable all the way up until it touches the ceiling.
2. A cable of density p pounds per foot is used to lift C pounds of coal up a mineshaft that is L feet deep. How much
work is done in this process?
3. A spring has a natural length of 20 cm. A 40 N force is required to stretch (and hold) the spring to a length of
30 cm. How much work is done in stretching the spring from 35 cm to 38 cm?
4. It requires a force of F Newtons to hold a spring stretched ` meters beyond its natural length. If L > `, how much
work, in terms of ` and L, is required to further stretch the spring from ` meters to L meters beyond the spring’s
natural length?
5. A tank in the shape of an inverted cone has a height of 15 meters and a base radius of 4 meters and is filled with
water to a depth of 12 meters. Determine the amount of work needed to pump all of the water to the top of the
tank. Assume that the density of the water is 1000 kg/m3 .
6. A tank in the shape of an inverted right circular cone that has a radius of r meters and a height of H meters is filled
to a height of h meters with a liquid of density D kilograms per cubic meter. If a spout is connected to the top of
the tank, how much work will it take to pump all of the liquid out of the tank through that spout?
7. A bucket of water of mass 30 kg is pulled at constant velocity up to a platform 50 meters above the ground. This
takes 20 minutes, during which time 4 kg of water drips out at a steady rate through a hole in the bottom. Find
the work needed to raise the bucket to the platform.
8. The average value of a function.
(a) Find the average of 21 , 13 , 14 , and 15 .
(b) Find the average value of f (x) = 1/x on the interval [2, 6]
(c) Can you visualize the first average as a Riemann sum approximation to the second average?
(d) Find a c in the interval [2, 6] such that f (c) is equal to the average value of f (x).
Review Problems Compute the following indefinite integrals, using u-substitution if necessary.
(a)
´ √
x x − 1dx
´ √
(b) x x2 − 1dx
ˆ
(c)
1
dx
4 + x2
(d)
´
ˆ
(e)
ˆ
(f)
√
x2 x − 1dx
1
√
dx
x x2 − 1
x
dx
4 + x2