Riemann Sums

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What is a Riemann Sum?

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A Riemann Sum is a method for approximating the total area under a curve by dividing it into smaller rectangles, calculating the area of each rectangle, and summing these areas.

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What is the formula for the Riemann Sum using right endpoints?

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What is the difference between left and right Riemann Sums?

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Left Riemann Sums use the left endpoints of the sub-intervals to calculate the height of the rectangles, while right Riemann Sums use the right endpoints.

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In the context of Riemann Sums, what does the term 'partition' refer to?

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A partition refers to the division of the interval into smaller sub-intervals for the purpose of approximating the area under the curve.

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What is the significance of the limit of Riemann Sums as n approaches infinity?

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The limit of Riemann Sums as n approaches infinity gives the exact value of the definite integral, representing the total area under the curve.

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If the function is increasing, how does the Riemann Sum compare to the actual integral using left endpoints?

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If the function is increasing, the left Riemann Sum will underestimate the actual value of the integral.

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What is the expression for the height of rectangles in a Riemann Sum using left endpoints?

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What is the relationship between Riemann Sums and definite integrals?

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Riemann Sums provide a method to approximate definite integrals, and as the number of sub-intervals increases, the Riemann Sum approaches the value of the definite integral.