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Questions and Answers
What is the formula for the left Riemann sum?
What is the purpose of the left Riemann sum?
What is the formula for the right Riemann sum?
What is the purpose of the right Riemann sum?
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What is the difference between the left and right Riemann sums?
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When would you use the left Riemann sum?
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What is Δx in the formula for the left Riemann sum?
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What is the variable 'n' in the formula for the right Riemann sum?
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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.
