Work performed
Pumping water from spherical tank
Calculate the work done pumping water out of a spherical tank of radius
Solution Divide the sphere of water into horizontal layers.
Coordinate
Work done pumping out water is constant across any single layer
Find formula for weight of a single layer.
Area of the layer at
Volume of the layer at
Weight of the layer is then
Plug in:
Find formula for vertical distance a given layer is lifted.
Layer at
Work per layer is the product.
Product of weight times height lifted:
Total work done is the integral.
Integrate over the layers:
Supplement: what if the spigot sits
above the tank?
Increase the height function from
Supplement: what if the tank starts at just
of water depth?
Integrate the water layers only: change bounds to
Water pumped from a frustum
Find the work required to pump water out of the frustum in the figure. Assume the weight of water is
Solution
Find weight of a horizontal slice.
Coordinate
Use
Linear decrease in
Area is
Find work to pump out a horizontal layer.
Layer at
Work to raise layer at
Integrate over all layers.
Integrate from top to bottom of frustum:
Final answer is
Raising a building
Find the work done to raise a cement columnar building of height
Solution
Divide the building into horizontal layers.
Work done raising up the layers is constant for each layer.
Find formula for weight of each layer.
Volume
Mass
Weight of layer
Find formula for work performed lifting one layer into place.
Work
Find total work as integral over the layers.
Total work
Raising a chain
An
Solution
Divide the chain into horizontal layers.
Each layer has vertical thickness
Each layer has weight
Find formula for distance each layer is raised.
Each layer is raised from
Compute total work.
Work to raise each layer is weight times distance raised:
Add the work over all layers:
Raising a leaky bucket
Suppose a bucket is hoisted by a cable up an
Solution
Convert to static format.
Compute rate of water weight loss per unit of vertical height:
Choose coordinate
Compute water weight at each height
Work formula.
Total work is integral of force times infinitesimal distance:
Integrate weight times
Plug in weight as force:
Compute integral:
Improper integrals
Improper integral - infinite bound
Show that the improper integral
Solution
Replace infinity with a new symbol
Compute the integral:
Take limit as
Find limit:
Apply definition of improper integral.
By definition:
Conclude that
Improper integral - infinite integrand
Show that the improper integral
Solution
Replace the
Compute the integral:
Take limit as
Find limit:
Apply definition of improper integral.
By definition:
Conclude that
Example - Improper integral - infinity inside the interval
Does the integral
Solution
It is tempting to compute the integral incorrectly, like this:
But this is wrong. There is an infinite integrand at
Identify infinite integrand at
.
Integral becomes:
Interpret improper integrals.
Limit interpretations:
Compute integrals.
Using
Take limits.
We have:
Neither limit is finite. For
Comparison to p-integrals
Determine whether the integral converges:
- (a)
- (b)
Solution
(a)
Integrand tends toward
for large .
Consider large
Try comparison to
Comparison attempt:
Validate. Notice
Apply comparison test.
We know:
We know:
We conclude:
(b)
Integrand tends toward
for large .
Consider large
Try comparison to
Comparison attempt:
Validate. Notice
Apply comparison test.
We know:
We know:
We conclude: