Riemann Summation and Left Riemann Sum
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Questions and Answers

What is the formula for the left Riemann sum?

  • $$L_n = \sum_{i=1}^{n} f(x_{i+1}) \Delta x$$
  • $$L_n = \sum_{i=1}^{n} f(x_{i-1}) \Delta x$$ (correct)
  • $$L_n = \sum_{i=1}^{n} f(x_i) \Delta x$$
  • $$L_n = \sum_{i=1}^{n} f(x_{i}) \Delta x^2$$

    • What is the purpose of the left Riemann sum?

  • To find the minimum value of a function
  • To calculate the exact area under a curve
  • To find the maximum value of a function
  • To approximate the area under a curve (correct)

    • What is the formula for the right Riemann sum?

  • $$R_n = \sum_{i=1}^{n} f(x_i) \Delta x$$ (correct)
  • $$R_n = \sum_{i=1}^{n} f(x_{i-1}) \Delta x$$
  • $$R_n = \sum_{i=1}^{n} f(x_{i}) \Delta x^2$$
  • $$R_n = \sum_{i=1}^{n} f(x_{i+1}) \Delta x$$

    • What is the purpose of the right Riemann sum?

    <p>To approximate the area under a curve</p> Signup and view all the answers

    What is the difference between the left and right Riemann sums?

    <p>The left Riemann sum uses the function value at the left endpoint of each subinterval, while the right Riemann sum uses the function value at the right endpoint.</p> Signup and view all the answers

    When would you use the left Riemann sum?

    <p>When you want to approximate the area under a curve</p> Signup and view all the answers

    What is Δx in the formula for the left Riemann sum?

    <p>The width of each subinterval</p> Signup and view all the answers

    What is the variable 'n' in the formula for the right Riemann sum?

    <p>The number of subintervals</p> Signup and view all the answers

    Study Notes

    Riemann Summation

    Riemann summation is a method used to approximate the total area under a curve by dividing the area into smaller rectangles and summing their areas.

    Left Riemann Sum

    • The left Riemann sum is a type of Riemann sum that approximates the area under a curve by using the left endpoint of each subinterval.
    • The formula for the left Riemann sum is:

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

    where: + $L_n$ is the left Riemann sum + $n$ is the number of subintervals + $f(x_{i-1})$ is the function value at the left endpoint of the $i^{th}$ subinterval + $\Delta x$ is the width of each subinterval

    Right Riemann Sum

    • The right Riemann sum is a type of Riemann sum that approximates the area under a curve by using the right endpoint of each subinterval.
    • The formula for the right Riemann sum is:

    $$R_n = \sum_{i=1}^{n} f(x_i) \Delta x$$

    where: + $R_n$ is the right Riemann sum + $n$ is the number of subintervals + $f(x_i)$ is the function value at the right endpoint of the $i^{th}$ subinterval + $\Delta x$ is the width of each subinterval

    Note: Both the left and right Riemann sums can be used to approximate the area under a curve, and the choice of which one to use depends on the specific problem and the desired level of accuracy.

    Riemann Summation

    • Riemann summation is a method to approximate the total area under a curve by dividing the area into smaller rectangles and summing their areas.

    Left Riemann Sum

    • Approximates the area under a curve using the left endpoint of each subinterval.
    • Formula: $$L_n = \sum_{i=1}^{n} f(x_{i-1}) \Delta x$$
    • Variables: $L_n$ (left Riemann sum), $n$ (number of subintervals), $f(x_{i-1})$ (function value at left endpoint of $i^{th}$ subinterval), $\Delta x$ (width of each subinterval)

    Right Riemann Sum

    • Approximates the area under a curve using the right endpoint of each subinterval.
    • Formula: $$R_n = \sum_{i=1}^{n} f(x_i) \Delta x$$
    • Variables: $R_n$ (right Riemann sum), $n$ (number of subintervals), $f(x_i)$ (function value at right endpoint of $i^{th}$ subinterval), $\Delta x$ (width of each subinterval)

    Comparison

    • Both left and right Riemann sums can be used to approximate the area under a curve.
    • The choice of which one to use depends on the specific problem and desired level of accuracy.

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    Description

    Learn about Riemann summation, a method to approximate the total area under a curve, and its type, Left Riemann Sum, with its formula and variables.

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