12/06/24 Riemann Sums

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What is a Riemann Sum?

Back

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.

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What does a Left Riemann Sum use to estimate area?

Back

It uses the left endpoints of sub-intervals to determine the height of rectangles.

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What does a Right Riemann Sum use to estimate area?

Back

It uses the right endpoints of sub-intervals to determine the height of rectangles.

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What is a Middle Riemann Sum?

Back

A method that uses the midpoints of sub-intervals to determine the height of rectangles for area estimation.

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What is a Trapezoidal Sum?

Back

A method that approximates the area under a curve by using trapezoids instead of rectangles.

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For a strictly increasing function, what does a Right Riemann Sum do?

Back

It overestimates the area under the curve.

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For a strictly decreasing function, what does a Right Riemann Sum do?

Back

It underestimates the area under the curve.

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How do you set up a Right Riemann Sum?

Back

Identify the right endpoints of each sub-interval and multiply by the width of the sub-intervals.

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What is the formula for calculating the area using a Riemann Sum?

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Area ≈ Σ (f(x_i) * Δx) where f(x_i) is the function value at the chosen points and Δx is the width of the sub-intervals.

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What is the significance of the number of sub-intervals in Riemann Sums?

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Increasing the number of sub-intervals generally leads to a more accurate approximation of the area.