224
4.2. RIEMANN SUMS
We read the symbol
100
k=1
k as “the sum from k equals 1 to 100 of k.” The variable k is
usually called the index of summation, and the letter that is used for this variable is
immaterial. Each sum in sigma notation involves a function of the index; for example,
10
k=1
(k
2 + 2k) = (1
2 + 2 · 1) + (2
2 + 2 · 2) + (3
2 + 2 · 3) + · · · + (10
2 + 2 · 10),
and more generally,
n
k=1
f (k) = f (1) + f (2) + · · · + f (n).
Sigma notation allows us the flexibility to easily vary the function being used to track the
pattern in the sum, as well as to adjust the number of terms in the sum simply by changing
the value of n. We test our understanding of this new notation in the following activity.
Activity 4.4.
For each sum written in sigma notation, write the sum long-hand and evaluate the
sum to find its value. For each sum written in expanded form, write the sum in sigma
notation.
(a)
5
k=1
(k
2 + 2)
(b)
6
i=3
(2i − 1)
(c) 3 + 7 + 11 + 15 + · · · + 27
(d) 4 + 8 + 16 + 32 + · · · + 256
(e)
6
i=1
1
2 i
⊳
Riemann Sums
When a moving body has a positive velocity function y = v(t) on a given interval [a, b],
we know that the area under the curve over the interval is the total distance the body
travels on [a, b]. While this is the fundamental motivating force behind our interest in
the area bounded by a function, we are also interested more generally in being able to
find the exact area bounded by y = f (x) on an interval [a, b], regardless of the meaning
or context of the function f . For now, we continue to focus on determining an accurate
estimate of this area through the use of a sum of the areas of rectangles, doing so in the
4.2. RIEMANN SUMS
We read the symbol
100
k=1
k as “the sum from k equals 1 to 100 of k.” The variable k is
usually called the index of summation, and the letter that is used for this variable is
immaterial. Each sum in sigma notation involves a function of the index; for example,
10
k=1
(k
2 + 2k) = (1
2 + 2 · 1) + (2
2 + 2 · 2) + (3
2 + 2 · 3) + · · · + (10
2 + 2 · 10),
and more generally,
n
k=1
f (k) = f (1) + f (2) + · · · + f (n).
Sigma notation allows us the flexibility to easily vary the function being used to track the
pattern in the sum, as well as to adjust the number of terms in the sum simply by changing
the value of n. We test our understanding of this new notation in the following activity.
Activity 4.4.
For each sum written in sigma notation, write the sum long-hand and evaluate the
sum to find its value. For each sum written in expanded form, write the sum in sigma
notation.
(a)
5
k=1
(k
2 + 2)
(b)
6
i=3
(2i − 1)
(c) 3 + 7 + 11 + 15 + · · · + 27
(d) 4 + 8 + 16 + 32 + · · · + 256
(e)
6
i=1
1
2 i
⊳
Riemann Sums
When a moving body has a positive velocity function y = v(t) on a given interval [a, b],
we know that the area under the curve over the interval is the total distance the body
travels on [a, b]. While this is the fundamental motivating force behind our interest in
the area bounded by a function, we are also interested more generally in being able to
find the exact area bounded by y = f (x) on an interval [a, b], regardless of the meaning
or context of the function f . For now, we continue to focus on determining an accurate
estimate of this area through the use of a sum of the areas of rectangles, doing so in the