Lesson 17 Riemann Sums
17.2 Lecture Notes
17.2.1 Estimating Area Using \(n\) Rectangles
To estimate the area under a curve \(f(x)\) on an interval \([a, b]\), divide the interval into \(n\) subintervals of equal width:
\[ \Delta x = \frac{b - a}{n} \]
Choose either left endpoints or right endpoints to evaluate the function:
- Left Endpoint Approximation (LRAM):
\[ A \approx \sum_{i=1}^{n} f(x_{i-1}) \Delta x \]
- Right Endpoint Approximation (RRAM):
\[ A \approx \sum_{i=1}^{n} f(x_i) \Delta x \]
Where:
\[ x_i = a + i\Delta x, \quad x_{i-1} = a + (i-1)\Delta x \]
These methods give a numerical estimate of the area under the curve.
17.2.2 Properties of Summation Notation
For any constants \(c\), and functions \(f(i)\), \(g(i)\):
- Linearity:
\[ \sum_{i=1}^{n} [f(i) + g(i)] = \sum_{i=1}^{n} f(i) + \sum_{i=1}^{n} g(i) \]
\[ \sum_{i=1}^{n} c \cdot f(i) = c \cdot \sum_{i=1}^{n} f(i) \]
- Additive Indexing:
\[ \sum_{i=m}^{n} f(i) = f(m) + f(m+1) + \dots + f(n) \]
17.2.3 Summation Formulas
These formulas are useful for simplifying Riemann sum expressions:
\[ \sum_{i=1}^{n} 1 = n \]
\[ \sum_{i=1}^{n} i = \frac{n(n+1)}{2} \]
\[ \sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6} \]
\[ \sum_{i=1}^{n} i^3 = \left( \frac{n(n+1)}{2} \right)^2 \]
17.2.4 Limit Definition of the Definite Integral
The definite integral of \(f(x)\) from \(a\) to \(b\) is defined as the limit of Riemann sums:
\[ \int_a^b f(x) \, dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x \]
Where: - \(\Delta x = \frac{b - a}{n}\) is the width of each subinterval, - \(x_i^*\) is any sample point in the \(i\)th subinterval \([x_{i-1}, x_i]\).
This definition captures the exact area under the curve as the number of rectangles \(n\) increases without bound and their width approaches zero.
17.3 Examples
Approximate the area under the curve \(y = x^3\) from \(x = 2\) to \(x = 5\) using a Right Endpoint approximation with 6 subdivisions.
Estimate the area under the graph of \(f(x) = x^2 + 6x + 12\) over the interval \([0, 2]\) using five approximating rectangles and right endpoints.
Evaluate the Riemann sum for \(f(x) = \ln(x) - 0.7\) over the interval \([1, 3]\) using four subintervals, taking the sample points to be right endpoints.
17.4 Practice Problems
Practice the techniques discussed in class and in the online videos by solving the following examples.
Use a Riemann sum to estimate the area under \(f(x) = x^2\) on [1,5] using \(n = 4\).
Use a Riemann sum to estimate the area under \(f(x) = 2x+3\) on [-2,3] using \(n=5\).
Use a Riemann sum to estimate the area under \(f(x) = \frac{1}{1+x}\) on [0,1] using \(n=4\).
Express in terms of \(n\):
\[\sum_{i=1}^{n}(i + i^{2})\]Express in terms of \(n\):
\[\sum_{i=1}^{n}(3i - 5 + i^{3})\]Express in terms of \(n\):
\[\frac{1}{n} \sum_{i=1}^{n} \left( \frac{2i}{3n} + \frac{3i^{3}}{2n^{3}} - \frac{4i^{2}}{7n^{2}} \right)\]Use the limit definition of the integral to evaluate
\[\int_{1}^{3} (2x+3) \, dx\]Use the limit definition of the integral to evaluate
\[\int_{0}^{3} (x + x^2) \, dx\](Applied) A race car has a velocity of
\[v(t) = 600(1 - e^{-0.5t})\]
ft/s, \(t\) seconds after starting. Use a Riemann sum with \(n = 10\) to estimate how far the car has traveled in the first 4 seconds.(Applied) The velocity of a stone moving under gravity \(t\) seconds after being thrown up at 4 m/s is given by
\[v(t) = -9.8t + 4\]
m/s. Use a Riemann sum with 5 subdivisions to estimate
\[\int_0^1 v(t) \, dt.\]
17.5 Self Assessment
Time yourself and try to solve the following questions within twenty minutes.
Use a Riemann sum to estimate the area under \(f(x) = 4-x^{2}\) on \([0,2]\) using \(n=4\).
Express in terms of \(n\):
\[\frac{2}{n}\sum_{i=1}^{\infty}(2-3i^{2})\]Evaluate:
\[\lim_{n \to \infty} \sum_{i=1}^{\infty} \left( \frac{i}{2n^{2}} - \frac{i^{2}}{3n^{3}} + \frac{4i^{3}}{n^{4}} \right)\]Use the limit definition of the integral to evaluate
\[\int_0^2 (4-x^{2}) \, dx\]The rate of U.S. per capita sales of bottled water for the period 2000–2008 can be approximated by
\[s(t) = 0.04 t^2 + 1.5 t + 17 \quad (0 \leq t \leq 8)\]
where \(t\) is the time in years since the start of 2000. Use a Riemann sum with \(n = 5\) to estimate the total U.S. per capita sales of bottled water from the start of 2003 to the start of 2008.
17.6 Lesson Checklist
This checklist is designed to help you keep track of what you need to work on. The main goal is to be aware of what you need to focus more attention on. Place an \(X\) in the appropriate box beside the skill below.
\[\begin{align*} &\textbf{Developing (D):} &&\textrm{You still need to work on this skill.}\\ &\textbf{Consistent (CON):} &&\textrm{You use the skill correctly most of the time.}\\ &\textbf{Competent (COM):} &&\textrm{You show mastery of the skill.} \end{align*}\]
| Skill | D | CON | COM |
|---|---|---|---|
| Estimate areas using approximating rectangles. | |||
| Manipulate expressions involving summation notation. | |||
| Use the limit definition of the integral. | |||
| Solve applied problems using Riemann sums. |