4.1: Riemann Sums

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    4.1 Riemann Sums

    Learning Objectives
    • Approximate the area under a curve by calculating the sum of rectangular regions, using an appropriate method such as left, right, or midpoint rectangles.
    • Calculate the area under a curve using Riemann sums by partitioning the interval into subintervals and applying the left, right, or midpoint approximation methods.

    Sigma Notation

    We anticipate dealing with sums of a large number of terms. To do so, we introduce sigma notation, named for the Greek letter \(\Sigma\text{,}\) which is the capital letter \(S\) in the Greek alphabet.

    For example, say we are interested in the sum of the first 100 natural numbers: \( 1 + 2 + 3 + \cdots + 100 \nonumber \).

    In sigma notation we write

    \[ \sum_{k=1}^{100} k = 1 + 2 + 3 + \cdots + 100\text{.} \nonumber \]

    We read the symbol \(\displaystyle \sum_{k=1}^{100} k\) as “the sum from \(k\) equals 1 to 100 of \(k\text{.}\)” The variable \(k\) is called the index of summation, and any letter can be used for this variable. The pattern in the terms of the sum is denoted by a function of the index; for example,

    \[ \sum_{k=1}^{10} (k^2 + 2k) = (1^2 + 2\cdot 1) + (2^2 + 2\cdot 2) + (3^2 + 2\cdot 3) + \cdots + (10^2 + 2\cdot 10)\text{,} \nonumber \]

    and more generally,

    \[ \sum_{k=1}^n f(k) = f(1) + f(2) + \cdots + f(n)\text{.} \nonumber \]

    Sigma notation allows us to easily vary the function being used to describe the terms in the sum, and to adjust the number of terms in the sum simply by changing the value of \(n\text{.}\) We will test our understanding of this new notation in the following activity.

    Riemann Sums

    Consider an arbitrary positive function \(f\) on \([a,b]\) with the interval subdivided as shown in the figure below.

    4_2_Interval.svg

    Figure \(\PageIndex{1}\). Subdividing the interval \([a,b]\) into \(n\) subintervals of equal length \(\Delta x\text{.}\) From Active Calculus by Matthew, Boelkins, licensed under CC BY.

    Let's use left endpoints. Then on each interval \([x_{i}, x_{i+1}]\text{,}\) the area of the rectangle formed is given by

    \[ A_{i+1} = f(x_i) \cdot \Delta x\text{,} \nonumber \]

    as seen in the figure below.

    4_2_LeftSum.svg

    Figure \(\PageIndex{2}\). Subdividing the interval \([a,b]\) into \(n\) subintervals of equal length \(\Delta x\) and

    approximating the area under \(y = f(x)\) over \([a,b]\) using left rectangles. From Active Calculus by Matthew, Boelkins, licensed under CC BY SA.

    If we let \(L_n\) denote the sum of the areas of these rectangles, we see that

    \begin{align*} L_n =\mathstrut & A_1 + A_2 + \cdots + A_{i+1} + \cdots + A_n\\[4pt] =\mathstrut & f(x_0) \cdot \Delta x + f(x_1) \cdot \Delta x + \cdots + f(x_i) \cdot \Delta x + \cdots + f(x_{n-1}) \cdot \Delta x\text{.} \end{align*}

    In the more compact sigma notation, we have

    \[ L_n = \sum_{i = 0}^{n-1} f(x_i) \Delta x\text{.} \nonumber \]

    Note that since the index of summation begins at \(0\) and ends at \(n-1\text{,}\) there are indeed \(n\) terms in this sum. We call \(L_n\) the left Riemann sum for the function \(f\) on the interval \([a,b]\text{.}\)

    To see how the Riemann sums for right endpoints and midpoints are constructed, consider Figure \(\PageIndex{3}\).

    4_2_RightMidSum.svg

    Figure \(\PageIndex{3}\). Riemann sums using right endpoints and midpoints. From Active Calculus by Matthew, Boelkins, licensed under CC BY SA.

    For the sum with right endpoints, we see that the area of the rectangle on an arbitrary interval \([x_i, x_{i+1}]\) is given by \(B_{i+1} = f(x_{i+1}) \cdot \Delta x\text{,}\) and that the sum of all such areas of rectangles is given by

    \begin{align*} R_n =\mathstrut & B_1 + B_2 + \cdots + B_{i+1} + \cdots + B_n\\[4pt] =\mathstrut & f(x_1) \cdot \Delta x + f(x_2) \cdot \Delta x + \cdots + f(x_{i+1}) \cdot \Delta x + \cdots + f(x_{n}) \cdot \Delta x\\[4pt] =\mathstrut & \sum_{i=1}^{n} f(x_i) \Delta x\text{.} \end{align*}

    We call \(R_n\) the right Riemann sum for the function \(f\) on the interval \([a,b]\text{.}\)

    For the sum that uses midpoints, we introduce the notation

    \[ \overline{x}_{i+1} = \frac{x_{i} + x_{i+1}}{2} \nonumber \]

    so that \(\overline{x}_{i+1}\) is the midpoint of the interval \([x_i, x_{i+1}]\text{.}\) For instance, for the rectangle with area \(C_1\) in Figure \(\PageIndex{3}\), we now have

    \[ C_1 = f(\overline{x}_1) \cdot \Delta x\text{.} \nonumber \]

    Hence, the sum of all the areas of rectangles that use midpoints is

    \begin{align*} M_n =\mathstrut & C_1 + C_2 + \cdots + C_{i+1} + \cdots + C_n\\[4pt] =\mathstrut & f(\overline{x_1}) \cdot \Delta x + f(\overline{x_2}) \cdot \Delta x + \cdots + f(\overline{x}_{i+1}) \cdot \Delta x + \cdots + f(\overline{x}_{n}) \cdot \Delta x\\[4pt] =\mathstrut & \sum_{i=1}^{n} f(\overline{x}_i) \Delta x\text{,} \end{align*}

    and we say that \(M_n\) is the middle Riemann sum for \(f\) on \([a,b]\text{.}\)

    Query\(\PageIndex{1}\)

    This page titled 4.1: Riemann Sums was last modified on Sun, 07 Dec 2025 01:10:58 GMT and is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Kevin Palencia.