368
6.4. PHYSICS APPLICATIONS: WORK, FORCE, AND PRESSURE
Hence, if we let △h tend to 0 and take the sum of all of the slices of work accomplished
on these small intervals, it follows that the total work is given by
W =
50
0
B(h) dh =
50
0
(12 + 8e
−0.1h ) dh.
While is a straightforward exercise to evaluate this integral exactly using the First Fundamental Theorem of Calculus, in applied settings such as this one we will typically use
computing technology to find accurate approximations of integrals that are of interest to
us. Here, it turns out that W =
50
0
(12 + 8e −0.1h ) dh ≈ 679.461 foot-pounds.
Our work in Preview Activity 6.1 and in the most recent example above employs the
following important general principle.
For an object being moved in the positive direction along an axis, x, by a force F(x),
the total work to move the object from a to b is given by
W =
b
a
F(x) dx.
Activity 6.10.
Consider the following situations in which a varying force accomplishes work.
(a) Suppose that a heavy rope hangs over the side of a cliff. The rope is 200 feet
long and weighs 0.3 pounds per foot; initially the rope is fully extended. How
much work is required to haul in the entire length of the rope? (Hint: set up
a function F(h) whose value is the weight of the rope remaining over the cliff
after h feet have been hauled in.)
(b) A leaky bucket is being hauled up from a 100 foot deep well. When lifted from
the water, the bucket and water together weigh 40 pounds. As the bucket is
being hauled upward at a constant rate, the bucket leaks water at a constant
rate so that it is losing weight at a rate of 0.1 pounds per foot. What function
B(h) tells the weight of the bucket after the bucket has been lifted h feet? What
is the total amount of work accomplished in lifting the bucket to the top of the
well?
(c) Now suppose that the bucket in (b) does not leak at a constant rate, but
rather that its weight at a height h feet above the water is given by B(h) =
25 + 15e −0.05h . What is the total work required to lift the bucket 100 feet? What
is the average force exerted on the bucket on the interval h = 0 to h = 100?
(d) From physics, Hooke’s Law for springs states that the amount of force required
to hold a spring that is compressed (or extended) to a particular length is
proportionate to the distance the spring is compressed (or extended) from its
6.4. PHYSICS APPLICATIONS: WORK, FORCE, AND PRESSURE
Hence, if we let △h tend to 0 and take the sum of all of the slices of work accomplished
on these small intervals, it follows that the total work is given by
W =
50
0
B(h) dh =
50
0
(12 + 8e
−0.1h ) dh.
While is a straightforward exercise to evaluate this integral exactly using the First Fundamental Theorem of Calculus, in applied settings such as this one we will typically use
computing technology to find accurate approximations of integrals that are of interest to
us. Here, it turns out that W =
50
0
(12 + 8e −0.1h ) dh ≈ 679.461 foot-pounds.
Our work in Preview Activity 6.1 and in the most recent example above employs the
following important general principle.
For an object being moved in the positive direction along an axis, x, by a force F(x),
the total work to move the object from a to b is given by
W =
b
a
F(x) dx.
Activity 6.10.
Consider the following situations in which a varying force accomplishes work.
(a) Suppose that a heavy rope hangs over the side of a cliff. The rope is 200 feet
long and weighs 0.3 pounds per foot; initially the rope is fully extended. How
much work is required to haul in the entire length of the rope? (Hint: set up
a function F(h) whose value is the weight of the rope remaining over the cliff
after h feet have been hauled in.)
(b) A leaky bucket is being hauled up from a 100 foot deep well. When lifted from
the water, the bucket and water together weigh 40 pounds. As the bucket is
being hauled upward at a constant rate, the bucket leaks water at a constant
rate so that it is losing weight at a rate of 0.1 pounds per foot. What function
B(h) tells the weight of the bucket after the bucket has been lifted h feet? What
is the total amount of work accomplished in lifting the bucket to the top of the
well?
(c) Now suppose that the bucket in (b) does not leak at a constant rate, but
rather that its weight at a height h feet above the water is given by B(h) =
25 + 15e −0.05h . What is the total work required to lift the bucket 100 feet? What
is the average force exerted on the bucket on the interval h = 0 to h = 100?
(d) From physics, Hooke’s Law for springs states that the amount of force required
to hold a spring that is compressed (or extended) to a particular length is
proportionate to the distance the spring is compressed (or extended) from its