Riemann Sum Flashcard

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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 sub-intervals and summing the areas of rectangles or trapezoids formed.

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What is a Left Riemann Sum?

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A Left Riemann Sum uses the left endpoints of sub-intervals to determine the height of rectangles, providing an approximation of the area under the curve.

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What is a Right Riemann Sum?

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A Right Riemann Sum uses the right endpoints of sub-intervals to determine the height of rectangles, providing an approximation of the area under the curve.

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What is a Midpoint Riemann Sum?

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A Midpoint Riemann Sum uses the midpoints of sub-intervals to determine the height of rectangles, providing a more accurate approximation of the area under the curve.

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What is a Trapezoidal Sum?

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A Trapezoidal Sum approximates the area under a curve by dividing it into trapezoids instead of rectangles, using the average of the left and right endpoints for the height.

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How do you calculate a Left Riemann Sum?

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To calculate a Left Riemann Sum, divide the interval into 'n' sub-intervals, find the left endpoint of each sub-interval, multiply the height (function value at left endpoint) by the width of the sub-interval, and sum these areas.

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How do you calculate a Right Riemann Sum?

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To calculate a Right Riemann Sum, divide the interval into 'n' sub-intervals, find the right endpoint of each sub-interval, multiply the height (function value at right endpoint) by the width of the sub-interval, and sum these areas.

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How do you calculate a Midpoint Riemann Sum?

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To calculate a Midpoint Riemann Sum, divide the interval into 'n' sub-intervals, find the midpoint of each sub-interval, multiply the height (function value at midpoint) by the width of the sub-interval, and sum these areas.

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What is the formula for the Trapezoidal Rule?

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The Trapezoidal Rule formula is: T = (b-a)/2n * (f(a) + 2 * Σf(x_i) + f(b)), where 'n' is the number of sub-intervals.

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What is the relationship between Riemann Sums and definite integrals?

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Riemann Sums are used to approximate the value of definite integrals, and as the number of sub-intervals increases, the Riemann Sum approaches the exact value of the integral.