6.3 Riemann Sums, Summation Notation, and Definite Integral Notation

• What are Riemann Sums used to approximate?

  Area under a curve
• The general form of a Riemann Sum is \sum_{i = 1}^{n} f(x_{i}^ * ) \Delta x</latex>
• In a Riemann Sum, $n$ represents the number of rectangles used to approximate the area under a curve.
• What does $\Delta x$ represent in a Riemann Sum?


  Width of each rectangle
• In a Riemann Sum, $x_{i}^ *$ is a point within each interval
• Left Riemann Sums use the left endpoint of each subinterval as the height of the rectangle.
• Match the Riemann Sum type with the method for determining the height of the rectangles:

  Left Riemann Sum ↔️ Left endpoints

  Right Riemann Sum ↔️ Right endpoints

  Midpoint Riemann Sum ↔️ Midpoints
• In a Left Riemann Sum, $x_{i}^ *$ is given by $a +$$(i - 1) \Delta x$
• What is the purpose of summation notation?

  Represent a sum of terms
• In summation notation, $i$ is called the index
• The upper limit of summation in summation notation is denoted by $\Delta x$.


  False
• What does the $expression$ represent in summation notation?


  Term being summed
• The summation \sum_{i = 1}^{5} i^{2}</latex> represents the sum $1^{2} +$$2^{2} +$$3^{2} +$$4^{2} +$$5^{2}$.
• How is summation notation used in the context of Riemann Sums?

  Sum the areas of rectangles
• Riemann Sums approximate the area under a curve by dividing the interval into smaller rectangles.
• The Midpoint Riemann Sum uses the midpoint of each subinterval to determine the height of the rectangles.
• What do Riemann Sums approximate?

  Area under a curve
• The width of each rectangle in a Riemann Sum is given by \Delta x
• What endpoint is used in a Left Riemann Sum to determine the height of the rectangles?

  Left
• Riemann Sums provide the exact area under a curve.

  False
• In summation notation, the lower limit of summation is denoted by lower
• What is the index of summation in the expression $\sum_{i = 1}^{5} i^{2}$?


  i
• The definite integral represents the limit of the Riemann Sum as the number of rectangles approaches infinity.
• What is the value of the definite integral $\int_{1}^{3} x^{2} dx$?


  $\frac{26}{3}$
• Match the Riemann Sum type with its $x_{i}^ *$ value:


  Left Riemann Sum ↔️ $a +$$(i - 1) \Delta x$

  Right Riemann Sum ↔️ $a +$$i \Delta x$

  Midpoint Riemann Sum ↔️ $a +$$(i - 0.5) \Delta x$
• As $n$ approaches infinity, the Riemann Sum converges to the exact value of the definite integral
• What is the value of $\Delta x$ in the Left Riemann Sum example for $\int_{0}^{2} x^{2} dx$ with $n =$$4$?


  $0.5$