Learning Outcomes

  1. Use sigma notation to represent summation.
  2. Use the sum of rectangular areas to approximate the area under a curve.
  3. Understand the notation of  Riemann Sums.

Textbook

  • Chapter 5.1  Approximating Areas   

Textbook Assignment

  • p. 523:    2, 12, 14-19 all

WeBWork Assignment

  • Integration-Riemann Sums

Exit problems of the session 

  1. 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  $[a, b]$ :
    1. Divide the interval  $[a, b]$  into  $n$  subintervals with equal length $\Delta x=\dfrac{b-a}{n}$;
    2. Let $x_i=a+i\cdot\Delta x$;
    3. 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

  1. $\rhd$ Summation notation and sums (4:26) Write $1+\cdots + 10$ and $1+\cdots + 100$ using summation notation.  Expand the sum $\displaystyle\sum_{i=0}^{50}\pi i^2$.
  2. * Summation notation A review with multiple-choice questions.
  3. $\rhd$ Worked examples: summation notation (5:20)
    • Consider the sum $2+5+8+11$. Find the expression that is equal to this sum.
    • Expand $\displaystyle\sum_{n=1}^{4}\dfrac{k}{n+1}$.
  4. * Practice: Summation notation. (4 problems)
  5. $\rhd$ Riemann approximation introduction (6:44) Approximate the area under the curve $y=x^2+1$ using rectangles.
  6. $\rhd$ 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 $x=4$.
  7. * Practice: Over and under estimation of Riemann sums. (4 problems)