6.3 Riemann Sums, Summation Notation, and Definite Integral Notation

• In Riemann Sums, the interval is divided into n subintervals of equal width.
• A larger value of n in Riemann Sums results in a more accurate approximation of the definite integral.


  True
• Steps for calculating a Riemann Sum

  1️⃣ Divide the interval into n subintervals

  2️⃣ Choose a sample point within each subinterval

  3️⃣ Calculate the function value at each sample point

  4️⃣ Multiply the function value by the subinterval width

  5️⃣ Sum the products
• The Midpoint Riemann Sum is often more accurate than Left-Hand or Right-Hand sums.
• What is the Left-Hand Riemann Sum approximation of \(f(x) = x^2\) over \([0, 2]\) with 4 subintervals?

  1.75
• What does the index variable in summation notation represent?

  Successive integer values
• What is the interval of integration for calculating Riemann sums in this example?

  [0, 2]
• What does the summation notation $\sum$ represent in the context of Riemann sums?


  Finite sum
• The summation notation $\sum$ is used to approximate the value of a definite integral.


  True
• In the summation expression $\sum_{i = 1}^{n} f(x_{i}) \Delta x$, what does `i` represent?


  Index variable
• The accuracy of a Riemann sum approximation increases as the number of subintervals increases.


  True
• A left-hand Riemann sum uses the left endpoint of each subinterval as the sample point.
• What sample point is used in a right-hand Riemann sum?

  Right endpoint
• The Midpoint Riemann Sum often provides more accurate approximations than Left-Hand or Right-Hand Riemann Sums because it uses the midpoint
• The index variable in summation notation takes on successive integer values.


  True
• What does the summation expression $\sum_{i = 1}^{n} f(x_{i}) \Delta x$ represent?


  Riemann sum
• The larger the number of subintervals in a Riemann sum, the closer the approximation is to the definite integral.


  True
• What happens to a constant factor in summation notation according to its properties?

  Pulled out
• A definite integral differs from an indefinite integral because it includes limits
• What is the antiderivative of $x^{2}$?


  $\frac{1}{3}x^{3}$
• Riemann sums are finite sums used to approximate definite integrals.


  True
• What formula approximates the definite integral using Riemann sums?

  \sum_{i = 1}^{n} f(x_{i}) \Delta x</latex>
• What happens to the accuracy of a Riemann sum as the number of subintervals increases?

  Increases
• The Riemann sum formula is written as \sum_{i = 1}^{n} f(x_{i}) \Delta x</latex>, where $x_{i}$ represents the sample point within each subinterval.
• Match the Riemann sum type with its formula:

  Left-Hand ↔️ $\sum_{i = 1}^{n} f(x_{i - 1}) \Delta x$

  Right-Hand ↔️ $\sum_{i = 1}^{n} f(x_{i}) \Delta x$

  Midpoint ↔️ \sum_{i = 1}^{n} f(\frac{x_{i - 1} + x_{i}}{2}) \Delta x
• In the summation notation $\sum_{i = a}^{b} f(i)$, the variable $a$ represents the start value of the index variable.
• What happens to a constant factor in summation notation?

  It is extracted
• $\sum_{i = 1}^{4} (3i + 2) =$$38$ illustrates the linearity and constant properties of summation notation.
• In the definite integral notation $\int_{a}^{b} f(x) \, dx$, the variable $a$ represents the lower limit of integration.


  True
• What does the definite integral notation $\int_{a}^{b} f(x) \, dx$ represent?


  Area under $f(x)$
• The definite integral notation $\int_{a}^{b} f(x) \, dx$ is used to calculate the area under the curve of $f(x)$ between $a$ and $b$.


  True
• In the definite integral notation $\int_{a}^{b} f(x) \, dx$, the function $f(x)$ is called the integrand
• How does a definite integral differ from an indefinite integral?

  Definite integral has limits
• What is the value of $\int_{0}^{2} x^{2} \, dx$?


  $\frac{8}{3}$
• As the number of subintervals in a Riemann sum increases, the approximation gets closer to the definite integral's value.


  True
• What is the value of $\Delta x$ in a left-hand Riemann sum for approximating $\int_{0}^{2} x^{2} \, dx$?


  $\frac{2}{n}$
• What do Riemann Sums approximate the value of?

  Definite integrals
• What is chosen within each subinterval to calculate a Riemann Sum?

  Sample point
• Match the type of Riemann Sum with its sample point:

  Left-hand ↔️ Left endpoint

  Right-hand ↔️ Right endpoint

  Midpoint ↔️ Midpoint
• What endpoint does a Left-Hand Riemann Sum use as its sample point?

  Left endpoint