6.2 Approximating Areas with Riemann Sums

• The formula for a Left Riemann Sum is \sum_{i = 1}^{n} f(x_{i - 1}) \Delta x
• The height of each rectangle in a Left Riemann Sum is determined by the function value at the left endpoint.


  True
• The width of each subinterval is calculated as \Delta x = \frac{b - a}{n}
• The Left Riemann Sum uses the function value at the right endpoint to determine the height of the rectangles.

  False
• To calculate a Left Riemann Sum, you first divide the interval into n equal subintervals.
• Steps to calculate the Left Riemann Sum for $f(x) =$$x^{2}$ on the interval [0, 2]</latex> with $n =$$4$


  1️⃣ Divide the interval into 4 subintervals of width $\Delta x =$$0.5$

  2️⃣ Determine the left endpoints: $0, 0.5, 1, 1.5$

  3️⃣ Evaluate the function at the left endpoints

  4️⃣ Calculate the Left Riemann Sum: $1.75$
• The Left Riemann Sum for f(x) = x^{2}</latex> on the interval $[0, 2]$ with $n =$$4$ is 1.75
• As the number of subintervals increases, the Left Riemann Sum approximation gets closer to the true area under the curve.


  True
• What is the Left Riemann Sum for $f(x) =$$x^{2}$ on $[0, 2]$ with 4 subintervals?


  1.75
• The Right Riemann Sum uses the right endpoint of each subinterval to determine the height of the rectangle.

  True
• To find the Right Riemann Sum for $f(x) =$$x^{2}$ on $[0, 2]$ with 4 subintervals, the width of each subinterval is 0.5
• The Left Riemann Sum and Right Riemann Sum both use the same endpoint to determine the height of the rectangle.

  False
• In a Right Riemann Sum, the height of the rectangle is determined by the function value at the right
• What is the Right Riemann Sum for $f(x) =$$x^{2}$ on $[0, 2]$ with $n =$$4$?


  3.75
• What point is used in a Left-Hand Sum to determine the height of the rectangle?

  Left endpoint
• What width is used in the formula for a Left Riemann Sum?

  Δx
• The Left Riemann Sum always overestimates the true area under the curve.

  False
• Match the type of Riemann Sum with its method of determining height:

  Left Riemann Sum ↔️ Left endpoint

  Right Riemann Sum ↔️ Right endpoint

  Midpoint Riemann Sum ↔️ Midpoint
• To calculate the area of each rectangle in a Left Riemann Sum, multiply the height by the width
• Steps to calculate a Left Riemann Sum to approximate the area under a curve

  1️⃣ Divide the interval into n equal subintervals

  2️⃣ Use the function value at the left endpoint of each subinterval to determine the height of the rectangle

  3️⃣ Multiply the height by the width of the subinterval to get the area of each rectangle

  4️⃣ Sum up the areas of all the rectangles
• As the number of subintervals n increases, the Left Riemann Sum approximation gets closer to the true area under the curve.


  True
• For the function f(x) = x^2</latex> on the interval [0, 2], the left endpoints with 4 subintervals are 0, 0.5, 1, and 1.5.
• Steps to calculate a Right Riemann Sum to approximate the area under a curve

  1️⃣ Divide the interval into n equal subintervals

  2️⃣ Use the function value at the right endpoint of each subinterval to determine the height of the rectangle

  3️⃣ Multiply the height by the width of the subinterval to get the area of each rectangle

  4️⃣ Sum up the areas of all the rectangles
• In a Left Riemann Sum, the height of the rectangle is determined by the left endpoint of each subinterval.

  True
• Match the feature with the correct type of Riemann Sum:

  Endpoint Used ↔️ Left Riemann Sum: Left ||| Right Riemann Sum: Right

  Height of Rectangle ↔️ Left Riemann Sum: $f(x_{i - 1})$ ||| Right Riemann Sum: $f(x_{i})$
• The height of the rectangle in a Left-Hand Sum is determined by the left endpoint of the subinterval.
• What are Riemann Sums used to approximate?

  Area under a curve
• As the number of subintervals increases, the Riemann Sum approximation becomes more accurate.

  True
• Riemann Sums are used to approximate the area under a curve
• As the number of subintervals increases, the Riemann Sum approximation gets closer to the true area under the curve.


  True
• In the Left Riemann Sum formula, `f(x_{i-1})` represents the function value at the left endpoint
• Steps to calculate a Left Riemann Sum

  1️⃣ Divide the interval into n equal subintervals

  2️⃣ Use the left endpoint of each subinterval to find the height of the rectangle

  3️⃣ Calculate the area of each rectangle

  4️⃣ Sum all the rectangle areas
• What width is used in the Left Riemann Sum to calculate the area of each rectangle?

  Δx
• The formula for the Left Riemann Sum includes summing the areas of all the rectangles.


  True
• What is the general formula for a Left Riemann Sum?

  $\sum_{i = 1}^{n} f(x_{i - 1}) \Delta x$
• What is the width of each subinterval when dividing the interval [0, 2] into 4 subintervals?

  $\Delta x =$$0.5$
• What is the Left Riemann Sum for $f(x) =$$x^{2}$ on [0, 2] with 4 subintervals?


  $1.75$
• What is the general formula for a Right Riemann Sum?

  $\sum_{i = 1}^{n} f(x_{i}) \Delta x$
• For the function $f(x) =$$x^{2}$ on the interval [0, 2], the right endpoints with 4 subintervals are 0.5, 1, 1.5, and 2.
• What is the Right Riemann Sum for $f(x) =$$x^{2}$ on [0, 2] with 4 subintervals?


  $3.75$