Riemann Sums:
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*There tends to be a free response question about Riemann Sums on the AP test, if there isn't a Riemann Sum there tends to be a Trapezoidal Sum problem
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Used to estimate the area under the curve using rectangles.
Left Riemann Sum:
*Make sure the distance between the points is equivalent when using this formula (If the points are different distances multiply b-a by each f(x))
A left Riemann Sum uses the points on the left to estimate the area under the curve. To calculate use the sum of all of the y values except the last point multiplied by the distance between the x values.
When the curve is decreasing the area found by a left Riemann Sum will be an overestimate.
When the curve is increasing the area found by a left Riemann Sum will be an underestimate. |
Right Riemann Sum:
*Make sure the distance between the points is equivalent when using this formula (If the points are different distances multiply b-a by each f(x))
A right Riemann Sum uses the points on the right to estimate the area under the curve. To calculate use the sum of all of the y values except the first point multiplied by the distance between the x values.
When the curve is increasing the area found by a right Riemann Sum will be an overestimate.
When the curve is decreasing the area found by a right Riemann Sum will be an underestimate. |
Midpoint Riemann Sum:
*Make sure the distance between the points is equivalent when using this formula (If the points are different distances multiply b-a by each f(x))
A midpoint Riemann Sum uses the average of the points to estimate the area under the curve. To calculate use the each of the averages of two y values next to each other multiplied by the distance between the values.
The midpoint Riemann Sum is a closer approximation than the left or right Riemann Sums.
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Trapezoidal Sum:
The trapezoidal sum uses the shapes of trapezoids to estimate the area under a curve. To calculate add up the y values, multiplying the middle y values by two, and multiply by half of the distance between x values.
The trapezoidal sum is a closer approximation than the Riemann Sums.
If the graph is concave up then the trapezoidal sum is an overestimate. If the graph is concave down then the trapezoidal sum is an underestimate. |
Trapezoidal Sum Error Theorem:
The trapezoidal sum error theorem tells you the maximum value that your answer from the trapezoidal sum can be off by.
Simpson's Rule: |
*The number of intervals must be even
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Simpson's Rule Error Theorem:
The Simpson's rule error theorem tells you the maximum value that your answer from the Simpson's rule can be off by.
Area of a Polar Curve:
Area between curves:
To find the area between two curves take the integral of the function on top and the function below, using the points of intersection as the integral endpoints.