4.2. RIEMANN SUMS
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among these sums is the location of the point at which the function is evaluated to
determine the height of the rectangle whose area is being computed in the sum. For
a left Riemann sum, we evaluate the function at the left endpoint of each subinterval,
while for right and middle sums, we use right endpoints and midpoints, respectively.
• The left, right, and middle Riemann sums are denoted L n , R n , and M n , with formulas
L n = f (x 0 )△x + f (x 1 )△x + · · · + f (x n−1 )△x =
n−1
i=0
f (x i )△x,
R n = f (x 1 )△x + f (x 2 )△x + · · · + f (x n )△x =
n
i=1
f (x i )△x,
M n = f (x 1 )△x + f (x 2 )△x + · · · + f (x n )△x =
n
i=1
f (x i )△x,
where x 0 = a, x i = a + i△x, and x n = b, using △x =
b−a
n . For the midpoint sum,
x i = (x i−1 + x i )/2.
Exercises
1. Consider the function f (x) = 3x + 4.
(a) Compute M 4 for y = f (x) on the interval [2, 5]. Be sure to clearly identify the
value of △x, as well as the locations of x 0 , x 1 , . . . , x 4 . Include a careful sketch
of the function and the corresponding rectangles being used in the sum.
(b) Use a familiar geometric formula to determine the exact value of the area of
the region bounded by y = f (x) and the x-axis on [2, 5].
(c) Explain why the values you computed in (a) and (b) turn out to be the same.
Will this be true if we use a number different than n = 4 and compute M n ? Will
L 4 or R 4 have the same value as the exact area of the region found in (b)?
(d) Describe the collection of functions g for which it will always be the case that
M n , regardless of the value of n, gives the exact net signed area bounded
between the function g and the x-axis on the interval [a, b].
2. Let S be the sum given by
S = ((1.4)
2 +1)·0.4+((1.8)
2 +1)·0.4+((2.2)
2 +1)·0.4+((2.6)
2 +1)·0.4+((3.0)
2 +1)·0.4.
(a) Assume that S is a right Riemann sum. For what function f and what interval
[a, b] is S an approximation of the area under f and above the x-axis on [a, b]?
Why?
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among these sums is the location of the point at which the function is evaluated to
determine the height of the rectangle whose area is being computed in the sum. For
a left Riemann sum, we evaluate the function at the left endpoint of each subinterval,
while for right and middle sums, we use right endpoints and midpoints, respectively.
• The left, right, and middle Riemann sums are denoted L n , R n , and M n , with formulas
L n = f (x 0 )△x + f (x 1 )△x + · · · + f (x n−1 )△x =
n−1
i=0
f (x i )△x,
R n = f (x 1 )△x + f (x 2 )△x + · · · + f (x n )△x =
n
i=1
f (x i )△x,
M n = f (x 1 )△x + f (x 2 )△x + · · · + f (x n )△x =
n
i=1
f (x i )△x,
where x 0 = a, x i = a + i△x, and x n = b, using △x =
b−a
n . For the midpoint sum,
x i = (x i−1 + x i )/2.
Exercises
1. Consider the function f (x) = 3x + 4.
(a) Compute M 4 for y = f (x) on the interval [2, 5]. Be sure to clearly identify the
value of △x, as well as the locations of x 0 , x 1 , . . . , x 4 . Include a careful sketch
of the function and the corresponding rectangles being used in the sum.
(b) Use a familiar geometric formula to determine the exact value of the area of
the region bounded by y = f (x) and the x-axis on [2, 5].
(c) Explain why the values you computed in (a) and (b) turn out to be the same.
Will this be true if we use a number different than n = 4 and compute M n ? Will
L 4 or R 4 have the same value as the exact area of the region found in (b)?
(d) Describe the collection of functions g for which it will always be the case that
M n , regardless of the value of n, gives the exact net signed area bounded
between the function g and the x-axis on the interval [a, b].
2. Let S be the sum given by
S = ((1.4)
2 +1)·0.4+((1.8)
2 +1)·0.4+((2.2)
2 +1)·0.4+((2.6)
2 +1)·0.4+((3.0)
2 +1)·0.4.
(a) Assume that S is a right Riemann sum. For what function f and what interval
[a, b] is S an approximation of the area under f and above the x-axis on [a, b]?
Why?