Week 8 recitation questions
A. Preamble
In lectures, we defined the average of a function f (x) that is integrable on [a, b] to be equal to
Z b
1
f (x) dx.
b−a a
Recall that the Mean Value Theorem for Integrals, which we applied in the proof of the Fundamental
Theorem of Calculus, states that if f (x) is not only integrable but continuous, the average value is attained;
Z b
1
f (x) dx. This prompted the following question.
that is, there exists c ∈ [a, b] such that f (c) =
b−a a
1. Come up with a discontinuous but integrable function f (x) whose average value on [a, b] is not attained.
Next, in our first of two applications, we considered centres of mass.
2. Consider a single point mass m at position x. What is the position of the centre of mass?
3. Consider two point masses m at positions x1 and x2 . What is the position of the centre of mass?
4. Consider two point masses m1 and m2 at positions x1 and x2 . What is the position of the centre of mass?
Our answers prompted the following definition. Given the area below the curve y = T (x), above the curve
y = B(x), and between x = a and x = b, and assuming an area density of ρ, the centre of mass has
x-coordinate
Z
Z
b
b
x (T (x) − B(x)) dx
ρ
a
Z
=
b
(T (x) − B(x)) dx
ρ
x (T (x) − B(x)) dx
ρ
a
M
,
a
where M is the total mass of the area, provided the integrals exist.
We applied the definition to the following question.
5. Let R be the quarter of the unit circle x2 + y 2 = 1 above the x-axis and to the right of the y-axis. Suppose
R is a disk of uniform density ρ. Find the x- and y-coordinates of its centre of mass.
Finally, we turned to our second of two applications, and solved the following differential equations.
dy
= 0.
dx
dy
7. Solve
= y.
dx
dy
8. Solve
= y, given the initial condition y(0) = 2.
dx
6. Solve
These are separable differential equations, as is the last one considered in lectures.
9. Solve
dy
= xy, given the initial condition y(0) = e.
dx
1
B. Questions
1. (a) Find the centre of mass of a semicircular plate of radius r of uniform density by considering the half
of the circle x2 + y 2 = r2 enclosed on the right half-plane.
(b) Apply the same principles used to define the x-coordinate of a centre of mass to define the y-coordinate
of a centre of mass.
(c) Use your definition to calculate the centre of mass of the plate in part (a), considering the half of the
circle x2 + y 2 = r2 above the x-axis.
2. (a) What is the centre of mass of a square of uniform density?
(b) Find the centre of mass of a plate of uniform density whose shape is a square of side length 2 directly
below a semicircle of radius 1.
3. Consider a plate of uniform density ρ whose shape is the area beneath the curve y = sin(x) from x = 0
to x = π2 .
(a) Find the mass of the plate.
(b) Find the x-coordinate of the centre of mass.
(c) Find the y-coordinate of the centre of mass.
4. Solve the following differential equations.
dy
y
(a)
= .
dx
x
dy
(b)
= y 2 sin(x).
dx
5. Solve the following differential equations with initial conditions.
log(x)
dy
=
, y(1) = 3.
dx
xy
dy
xy
(b)
= 2
, y(0) = 2.
dx
x +1
6. Find an infinite number of curves that intersect at right angles all ellipses of the form x2 + 2y 2 = a2 .
(a)
2