Learning Outcomes
- Use sigma notation to represent summation.
- Use the sum of rectangular areas to approximate the area under a curve.
- Understand the notation of Riemann Sums.
Textbook
- Chapter 5.1 Approximating Areas
Textbook Assignment
- p. 523: 2, 12, 14-17 all
WeBWork Assignment
- Integration-Riemann Sums
Exit problems of the session
-
Compute $L_4$ and $R_4$ to approximate the area under the curve $f(x)=x^2+1$ on $[0, 4]$ . Explain whether it is an overestimation or underestimation.
Key Concepts
- Sigma notation of the form $\displaystyle\sum_{i=1}^{n}a_i$ is useful to express long sums of values in compact form.
- Riemann Sums are expressions of the form $\displaystyle\sum_{i=1}^{n}f(x^*_i)\Delta x$, and can be used to estimate the area under the curve $y=f(x)$.
- Approximate the area under the curve $y=f(x)$ on the interval $[a, b]$ :
- Divide the interval $[a, b]$ into $n$ subintervals with equal length $\Delta x=\dfrac{b-a}{n}$;
- Let $x_i=a+i\cdot\Delta x$;
- We can either use Left Endpoint Approximation:
$A\approx L_n=f(x_0)\Delta x+f(x_1)\Delta x+\cdots+f(x_{n-1})\Delta x=\displaystyle\sum_{i=1}^{n}f(x_{i-1})\Delta x$
or Right Endpoint Approximation:
$A\approx R_n=f(x_1)\Delta x+f(x_2)\Delta x+\cdots+f(x_{n})\Delta x=\displaystyle\sum_{i=1}^{n}f(x_i)\Delta x$
Videos and Practice Problems of Selected Topics
- $\rhd$ Summation notation and sums (4:26) Write $1+\cdots + 10$ and $1+\cdots + 100$ using summation notation. Expand the sum $\displaystyle\sum_{i=0}^{50}\pi i^2$.
- * Summation notation A review with multiple-choice questions.
- $\rhd$ Worked examples: summation notation (5:20)
- Consider the sum $2+5+8+11$. Find the expression that is equal to this sum.
- Expand $\displaystyle\sum_{n=1}^{4}\dfrac{k}{n+1}$.
- * Practice: Summation notation. (4 problems)
- $\rhd$ Riemann approximation introduction (6:44) Approximate the area under the curve $y=x^2+1$ using rectangles.
- $\rhd$ Left and right Riemann sums (4:00) Given the graph of a function $y=g(x)$, find the left and right Riemann sums that approximate the area under $y=g(x)$ between $x=2$ and $x=4$.
- * Practice: Over and under estimation of Riemann sums. (4 problems)
STEM Application
MAT1475 Optimization Assignment