Area, displacement and Riemann sums

Section 5.2 Area, displacement and Riemann sums

We now go back to the beginning. In our preview of calculus (Section 2.1), we saw that we can estimate the area under a curve by replacing it with a finite number of rectangles of appropriate heights and widths. In this section we study this approach more rigourously; it gives rise to the concept of “Riemann sums”. We introduce the “summation notation”, study some of its properties, and use it to calculate a few simple Riemann sums. We also explore how taking the limit of a Riemann sum where the number of rectangles becomes infinite (in which case their widths go to zero) gives rise to a precise definition of the area under a curve.

Objectives

You should be able to:

Describe and illustrate how to approximate the area under a curve using approximating rectangles and a Riemann sum.
Construct a Riemann sum to approximate the area under the curve of a given function over a given interval \([a,b]\) using \(n\) subintervals, with either left endpoints, right endpoints, or mid endpoints.
Calculate the value of a Riemann sum for a given function over a given interval for a given value of \(n\text{.}\)
Describe the limit process that arises in the calculation of the precise area under a curve using Riemann sums.
Calculate the area under a curve using limits of Riemann sums.
Evaluate simple finite sums using the summation notation.

Subsection 5.2.1 Instructional video

Subsection 5.2.2 Key concepts

Concept 5.2.1. Summation notation.

For \(\{a_i\}\) a set of numbers indexed by the integers, and for integers \(k\leq n\) we have

\begin{equation*} \sum_{i=k}^{n}a_i=a_{k}+a_{{k+1}}+\dots+a_{{n-1}}+a_{n}. \end{equation*}

Concept 5.2.2. Properties of sums.

\begin{equation*} \sum_{i=k}^{n}c a_i=c \sum_{i=k}^{n}a_i, \qquad\qquad \sum_{i=k}^{n}(a_i\pm b_i)=\sum_{i=k}^{n}a_i\pm \sum_{i=k}^{n}b_i, \end{equation*}

and for any integer \(k\) with \(a \lt k \lt b\text{,}\)

\begin{equation*} \sum_{i=a}^{b}c_i=\sum_{i=a}^{k}c_i + \sum_{i=k+1}^{b}c_i. \end{equation*}

Concept 5.2.3. Four useful finite sums.

\begin{align*} \sum_{i=1}^{n}1\amp =n,\amp \sum_{i=1}^{n}i^2 \amp=\dfrac{n(n+1)(2n+1)}{6},\\ \sum_{i=1}^{n}i \amp=\dfrac{n(n+1)}{2}, \amp \sum_{i=1}^{n}i^3\amp=\left(\dfrac{n(n+1)}{2}\right)^2. \end{align*}

Concept 5.2.4. Riemann sums.

For \(f(x)\geq 0\) on the interval \([a,b]\text{,}\) to calculate the approximate area bounded by \(y=f(x)\text{,}\) \(y=0\text{,}\) \(x=a\) and \(x=b\) using \(n\) intervals of equal width and the right endpoints of each interval we define the Riemann sum:

\begin{equation*} R_n=\sum_{i=1}^{n}\Delta x\cdot f(a+i\Delta x),\qquad\text{ where }\,\,\Delta x=(b-a)/n. \end{equation*}

When \(n\) is increased the number of rectangles used is increased and the approximation is improved.

Concept 5.2.5. Right endpoints, left endpoints, and mid endpoints.

This is the Riemann sum for “right endpoints”, meaning that the rectangles approximating the area under the curve have heights \(f(x_i)\) with \(x_i\) corresponding to the right endpoints of the rectangles. We can also define similarly Riemann sums \(L_n\) for “left endpoints”, and \(M_n\) for “mid endpoints”.

Concept 5.2.6. Limit of infinite number of rectangles of zero width.

If we let \(n \to \infty\text{,}\) in whic case \(\Delta x \to 0\text{,}\) then the limit of all the Riemann sums \(R_n\text{,}\) \(L_n\text{,}\) and \(M_n\) (right endpoint, left endpoint, and mid endpoint) are all equal:

\begin{equation*} A = \lim_{n \to \infty} R_n = \lim_{n \to \infty} L_n = \lim_{n \to \infty} M_n. \end{equation*}

This limit defines the exact area under the curve \(A\) .

Concept 5.2.7. Interpretation in kinematics.

If \(f(t) \geq 0\) is a velocity function then the Riemann sum \(R_n\) (or \(L_n\) or \(M_n\)) can be used to approximate the distance traveled during the interval of time from \(t=a\) to \(t=b\text{,}\) and \(\displaystyle \lim_{n \to \infty} R_n\) becomes the exact distance traveled.

MATH 144: Calculus for the Physical Sciences I