4.2. RIEMANN SUMS
223
(a) How are the heights of rectangles in the left-most diagram being chosen? Explain,
and hence determine the value of
S = A 1 + A 2 + A 3 + A 4
by evaluating the function y = v(t) at appropriately chosen values and observing
the width of each rectangle. Note, for example, that
A 3 = v(1) ·
1
2
= 2 ·
1
2
= 1.
(b) Explain how the heights of rectangles are being chosen in the middle diagram and
find the value of
T = B 1 + B 2 + B 3 + B 4 .
(c) Likewise, determine the pattern of how heights of rectangles are chosen in the
right-most diagram and determine
U = C 1 + C 2 + C 3 + C 4 .
(d) Of the estimates S, T, and U, which do you think is the best approximation of D,
the total distance the person traveled on [0, 2]? Why?
⊲⊳
Sigma Notation
It is apparent from several different problems we have considered that sums of areas of
rectangles is one of the main ways to approximate the area under a curve over a given
interval. Intuitively, we expect that using a larger number of thinner rectangles will provide
a way to improve the estimates we are computing. As such, we anticipate dealing with
sums with a large number of terms. To do so, we introduce the use of so-called sigma
notation, named for the Greek letter Σ, which is the capital letter S in the Greek alphabet.
For example, say we are interested in the sum
1 + 2 + 3 + · · · + 100,
which is the sum of the first 100 natural numbers. Sigma notation provides a shorthand
notation that recognizes the general pattern in the terms of the sum. It is equivalent to
write
100
k=1
k = 1 + 2 + 3 + · · · + 100.
223
(a) How are the heights of rectangles in the left-most diagram being chosen? Explain,
and hence determine the value of
S = A 1 + A 2 + A 3 + A 4
by evaluating the function y = v(t) at appropriately chosen values and observing
the width of each rectangle. Note, for example, that
A 3 = v(1) ·
1
2
= 2 ·
1
2
= 1.
(b) Explain how the heights of rectangles are being chosen in the middle diagram and
find the value of
T = B 1 + B 2 + B 3 + B 4 .
(c) Likewise, determine the pattern of how heights of rectangles are chosen in the
right-most diagram and determine
U = C 1 + C 2 + C 3 + C 4 .
(d) Of the estimates S, T, and U, which do you think is the best approximation of D,
the total distance the person traveled on [0, 2]? Why?
⊲⊳
Sigma Notation
It is apparent from several different problems we have considered that sums of areas of
rectangles is one of the main ways to approximate the area under a curve over a given
interval. Intuitively, we expect that using a larger number of thinner rectangles will provide
a way to improve the estimates we are computing. As such, we anticipate dealing with
sums with a large number of terms. To do so, we introduce the use of so-called sigma
notation, named for the Greek letter Σ, which is the capital letter S in the Greek alphabet.
For example, say we are interested in the sum
1 + 2 + 3 + · · · + 100,
which is the sum of the first 100 natural numbers. Sigma notation provides a shorthand
notation that recognizes the general pattern in the terms of the sum. It is equivalent to
write
100
k=1
k = 1 + 2 + 3 + · · · + 100.