Riemann sum

Lessons

• Introduction
  Riemann Sum overview:
  a)

  area under the curve
  
  
  b)

  summation formula
  
  
  c)

  what is "Riemann Sum"?
  
  
  d)

  integrals expressed as Riemann Sum
  
  

---

• 1.
  Finding the area under the graph of a function using a graphing calculator.
  Consider the function $f\left( x \right) = {x^2}$, $1 \le x \le 3$.
  Find the area under the graph of $f$ using a graphing calculator.

---

• 2.
  Finding the area under the graph of a function using the Riemann Sum.
  Consider the function $f\left( x \right) = {x^2}$, $1 \le x \le 3$.
  Estimate the area under the graph of $f$ using four approximating rectangles and taking the sample points to be:
  a)

  right endpoints
  
  
  b)

  left endpoints
  
  
  c)

  midpoints
  
  

---

• 3.
  Evaluating integrals with a Riemann Sum
  Consider the function $f\left( x \right) = {x^2} - 5x + 3$, $2 \le x \le 5$.
  a)

  Evaluate the Right Riemann sum for $f$ with 6 sub-intervals.
  
  
  b)

  Evaluate $\int_2^5 \left( {{x^2} - 5x + 3} \right)dx$, by finding the Riemann sum for $f$ with $\infty$ intervals.
  
  

---

• 4.
  Evaluating Riemann Sum with trapezoids
  Consider the function $f\left( x \right) = {x^2}$, $1 \le x \le 3$. Estimate the area under the graph of $f$ using four approximating trapezoids.

---