6.3. DENSITY, MASS, AND CENTER OF MASS
359
and for a quantity being measured by a function f on an interval [a, b], the average value
of the quantity on [a, b] is
1
b − a
b
a
f (x) dx.
As we continue to think about problems involving the distribution of mass, it is natural
to consider the idea of a weighted average, where certain quantities involved are counted
more in the average.
A common use of weighted averages is in the computation of a student’s GPA, where
grades are weighted according to credit hours. Let’s consider the scenario in Table 6.1.
If all of the classes were of the same weight (i.e., the same number of credits), the
student’s GPA would simply be calculated by taking the average
3.3 + 3.7 + 2.7 + 2.7
4
= 3.1.
But since the chemistry and calculus courses have higher weights (of 5 and 4 credits
respectively), we actually compute the GPA according to the weighted average
3.3 · 5 + 3.7 · 4 + 2.7 · 3 + 2.7 · 3
5 + 4 + 3 + 3
= 3.16.
The weighted average reflects the fact that chemistry and calculus, as courses with higher
credits, have a greater impact on the students’ grade point average. Note particularly that
in the weighted average, each grade gets multiplied by its weight, and we divide by the
sum of the weights.
In the following activity, we explore further how weighted averages can be used to find
the balancing point of a physical system.
Activity 6.8.
For quantities of equal weight, such as two children on a teeter-totter, the balancing
point is found by taking the average of their locations. When the weights of the
quantities differ, we use a weighted average of their respective locations to find the
balancing point.
(a) Suppose that a shelf is 6 feet long, with its left end situated at x = 0. If one
book of weight 1 lb is placed at x 1 = 0, and another book of weight 1 lb is
placed at x 2 = 6, what is the location of x, the point at which the shelf would
(theoretically) balance on a fulcrum?
(b) Now, say that we place four books on the shelf, each weighing 1 lb: at x 1 = 0,
at x 2 = 2, at x 3 = 4, and at x 4 = 6. Find x, the balancing point of the shelf.
(c) How does x change if we change the location of the third book? Say the
locations of the 1-lb books are x 1 = 0, x 2 = 2, x 3 = 3, and x 4 = 6.
359
and for a quantity being measured by a function f on an interval [a, b], the average value
of the quantity on [a, b] is
1
b − a
b
a
f (x) dx.
As we continue to think about problems involving the distribution of mass, it is natural
to consider the idea of a weighted average, where certain quantities involved are counted
more in the average.
A common use of weighted averages is in the computation of a student’s GPA, where
grades are weighted according to credit hours. Let’s consider the scenario in Table 6.1.
If all of the classes were of the same weight (i.e., the same number of credits), the
student’s GPA would simply be calculated by taking the average
3.3 + 3.7 + 2.7 + 2.7
4
= 3.1.
But since the chemistry and calculus courses have higher weights (of 5 and 4 credits
respectively), we actually compute the GPA according to the weighted average
3.3 · 5 + 3.7 · 4 + 2.7 · 3 + 2.7 · 3
5 + 4 + 3 + 3
= 3.16.
The weighted average reflects the fact that chemistry and calculus, as courses with higher
credits, have a greater impact on the students’ grade point average. Note particularly that
in the weighted average, each grade gets multiplied by its weight, and we divide by the
sum of the weights.
In the following activity, we explore further how weighted averages can be used to find
the balancing point of a physical system.
Activity 6.8.
For quantities of equal weight, such as two children on a teeter-totter, the balancing
point is found by taking the average of their locations. When the weights of the
quantities differ, we use a weighted average of their respective locations to find the
balancing point.
(a) Suppose that a shelf is 6 feet long, with its left end situated at x = 0. If one
book of weight 1 lb is placed at x 1 = 0, and another book of weight 1 lb is
placed at x 2 = 6, what is the location of x, the point at which the shelf would
(theoretically) balance on a fulcrum?
(b) Now, say that we place four books on the shelf, each weighing 1 lb: at x 1 = 0,
at x 2 = 2, at x 3 = 4, and at x 4 = 6. Find x, the balancing point of the shelf.
(c) How does x change if we change the location of the third book? Say the
locations of the 1-lb books are x 1 = 0, x 2 = 2, x 3 = 3, and x 4 = 6.