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, 14-17 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)

STEM Application

MAT1475 Optimization Assignment