Math 1311 Lab, Fall 2015
TA: Matteo Altavilla
Name:
uNID:
Worksheet #10
(30 + 5 points)
Worksheet #10 is due Thursday, November 19th . You are encouraged to work with other people
to solve these problems, but you have to write the solutions down individually. Please turn in these
sheets only. You can write at the bottom of each page and/or on the back. Show all the steps
and explain your reasoning when necessary: correct solutions with no explanation do not count as
complete. Partial credit will be assigned to incomplete answers.
Archimedes of Syracuse (287 BC - 212 BC) was an Ancient Greek mathematician, physicist, engineer,
inventor and astronomer. He was an awesome guy! He definitely was one of the best mathematicians
of ancient times, and he discovered a lot of stuff about areas, volumes and lengths of curves; for
example, he found the formula for the volume of a sphere and computed a lot of exact decimal digits
of π. The amazing part is that he did not know about integrals at all, but he was able to carry
out very precise computations using methods similar to the approximation by Riemann sums. The
problems on this Worksheet are inspired by his discoveries, but only cover the mathematical aspects
of his work. As future engineers, I suggest you to read something about all the great stuff he invented
and studied (Archimedes’ principle for fluids, the Archimedes screw, the Heath Ray, levers); here’s
also a picture of him.
Problem 1. (10 points) One of Archimedes’ most famous results in mathematics is the formula for
the area of a parabolic segment. A parabolic segment is the portion of the plane enclosed between a
parabola and a secant line. Archimedes’ formula says that the area of a parabolic segment is equal to
2/3 of the area of the rectangle in which it is inscribed. Such a rectangle is formed by the secant line
itself, a parallel line to it which is tangent to the parabola, and the two perpendicular lines passing
through the secancy points (see figure below for an example).
Given the parabola y = x2 −5x+11, and the horizontal line y = 5, find the area of the corresponding
parabolic segment. Then, verify Archimedes’ formula in this special case. [Warning: the parabola in
the picture above is not the one I’m asking you to consider].
Problem 2. (10+5 points) Archimedes’ most important discovery is the formula to compute the
volume of a sphere. He himself was so proud of this formula that he asked for it to be engraved on
his tomb (remember that he did not know integrals!). You saw the proof of the formula in class, so
for this problem we will focus on other shapes, using the same ideas.
(a) Prove that the volume of a cone of base radius r and height h is given by V = 13 πr2 h.
(b) In solving the previuos item (and also in class) you had to use the formula for the area of a
circle at some point. Prove that the area of a circle of radius r is given by πr2 .
(c) Bonus/Hint. To prove item (b) you need the formula for the length of a circumference. Prove
that the length of a circumference of radius r is 2πr.
Problem 3. (10 points) Archimedes is known also for his extended study on spirals. The so-called
Archimedean spiral is a plane curve of parametric equations
x(t) = t cos t
s(t) =
t ≥ 0.
y(t) = t sin t
This curve is one of the first examples of a mathematical object arising from a physical setting: it
represents the motion of an object getting further and further from a point while staying on a straight
line which in turn rotates around the origin. The shape of this spiral is what you get for instance
when you wind up a rope, and it is used to build gears such as the ones that are inside a mechanical
clock.
(a) Write down the integral representing the arc length of the spiral for t that goes from 0 to 6π.
How many times does the spiral wind up on itself in this case?
(b) Use a Computer Algebra System to find the exact value of the integral from item (a), and then
use a method of your choice to find an approximation of it (you can use rectangles, or some
fancier method from Section 5.9 of your textbook).