What is the definition of a Riemann Sum?
Riemann Sum Notation - Pugh

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Mathematics
•
12th Grade
•
Hard
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1.
FLASHCARD QUESTION
Front
Back
A Riemann Sum is a method for approximating the total area under a curve by dividing it into rectangles, calculating the area of each rectangle, and summing these areas.
2.
FLASHCARD QUESTION
Front
What does the notation \( \lim_{n\rightarrow\infty} \sum_{k=1}^n f(x_k^*) \Delta x \) represent?
Back
This notation represents the definite integral of a function \( f \) from a to b, where \( x_k^* \) is a sample point in each subinterval and \( \Delta x \) is the width of each subinterval.
3.
FLASHCARD QUESTION
Front
What is the lower limit in the context of Riemann Sums?
Back
The lower limit is the starting point of the interval over which the function is being integrated.
4.
FLASHCARD QUESTION
Front
What does \( a + \frac{b-a}{n}k \) represent in summation notation?
Back
It represents the x-value at the k-th subinterval in the partition of the interval [a, b].
5.
FLASHCARD QUESTION
Front
How do you determine the width of each rectangle in a Riemann Sum?
Back
The width of each rectangle is determined by \( \Delta x = \frac{b-a}{n} \), where \( n \) is the number of rectangles.
6.
FLASHCARD QUESTION
Front
What is the significance of the upper limit in Riemann Sums?
Back
The upper limit is the endpoint of the interval over which the function is being integrated.
7.
FLASHCARD QUESTION
Front
What is the integral notation for the summation \( \lim_{n\rightarrow\infty}\sum_{k=1}^n\left(\frac{3}{n}\right)\left[\left(\frac{3k}{n}\right)^3\right] \)?
Back
The integral notation is \( \int_0^3 x^3 dx \).
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