1. Applications of Integration
Section 7.5a

2. Work Done Lifting
A leaky bucket weighs 22 N empty. It is lifted from the
ground at a constant rate to a point 20m above the
ground by a rope weighing 0.4 N/m. The bucket starts
with 70 N (approx. 7.1 L) of water, but it leaks at a
constant rate and just finishes draining as the bucket
reaches the top. Find the amount of work done
(a) lifting the bucket alone;
(b) lifting the water alone;
(c) lifting the rope alone;
(d) lifting the bucket, water, and rope together.
Remember that work is the area under the force graph…

3. Work Done Lifting (a) lifting the bucket alone
The bucket’s weight is constant, so we need to exert
the 22 N force along the entire 20m distance: F
22N
Work
20m x

4. Work Done Lifting (b) lifting the water alone
The force needed to lift the water is equal to its
weight, which decreases steadily from 70 N to 0 N
over the 20m lift: F
70N
Work
20m x

5. Work Done Lifting (c) lifting the rope alone
The force to lift the rope also varies, starting at 8 N
and ending at 0 N: F 8N
Work
20m x

6. Work Done Lifting (d) lifting the bucket, water, and rope together
Simply sum for the total work done:

7. Work Done Pumping
The conical tank shown is filled to within 2 ft of the
top with olive oil weighing How much work
does it take to pump the oil to the rim of the tank? y
Volume of one of
these thin slabs: x

8. Work Done Pumping
The conical tank shown is filled to within 2 ft of the
top with olive oil weighing How much work
does it take to pump the oil to the rim of the tank?
The force to lift this slab is equal to its weight:
This force must be applied over a distance of
To calculate the work of moving all of the slabs, use
a Riemann sum from y = 0 to y = 8…

9. Work Done Pumping
The conical tank shown is filled to within 2 ft of the
top with olive oil weighing How much work
does it take to pump the oil to the rim of the tank?

10. Hooke’s Law Revisited A bathroom scale is compressed 1/16 in. when a
150-lb person stands on it. Assuming the scale
behaves like a spring that obeys Hooke’s Law.
How much does someone who compresses the
scale 1/8 in. weigh?
with ,

11. Hooke’s Law Revisited A bathroom scale is compressed 1/16 in. when a
150-lb person stands on it. Assuming the scale
behaves like a spring that obeys Hooke’s Law.
(b) How much work is done in compressing the scale
1/8 in.?