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Riemann Sum
An approximation of the integral of a function over an interval by dividing it into smaller subintervals and summing the areas of rectangles.
Interval [a,b]
The range over which the function is being integrated using Riemann Sums.
Δx
The width of each subinterval in a Riemann Sum, calculated as (b-a)/n.
Left Riemann Sum
Approximates the area under the curve by using the left endpoints of subintervals to determine rectangle heights.
Right Riemann Sum
Approximates the area under the curve by using the right endpoints of subintervals to determine rectangle heights.
Midpoint Riemann Sum
Uses the midpoints of the subintervals to determine the heights of the rectangles for approximation.
Trapezoidal Rule
A method that approximates the area under a curve by dividing it into trapezoids instead of rectangles.
Definite Integral
The limit of a Riemann Sum as the width of the subintervals approaches zero, representing the exact area under the curve.
Integral of a function
The total accumulation of a function's values over an interval, represented graphically as the area under the curve.
Approximation Techniques
Methods used to estimate the value of a definite integral, including various Riemann Sums.
Fundamental Theorem of Calculus
Establishes the relationship between differentiation and integration, showing how to compute definite integrals.
Numerical Integration
Approaches for calculating the integral of a function when an analytical solution is difficult or impossible.
Error and Bounds
The difference between the actual value of an integral and the value estimated by Riemann Sums, often expressed as upper and lower bounds.
AP Calculus
A standardized Advanced Placement course assessing understanding of fundamental calculus concepts, including Riemann Sums.
Visualization
The process of representing mathematical concepts graphically, helping to understand how Riemann Sums approximate areas under curves.
Sums for Larger n
Refers to the practice of calculating Riemann Sums with a greater number of subintervals for greater accuracy.
Height of Rectangles
In Riemann Sums, determined by either left endpoints, right endpoints, midpoints, or through trapezoidal approximation.
Area Under a Curve
The total space enclosed between the curve of a function and the horizontal axis, obtainable through integration.
Subintervals
Smaller divisions of the main interval [a,b] used to approximate the area with rectangles in a Riemann Sum.
Calculator programs
Tools or applications used to assist in evaluating Riemann Sums, especially for large n values.
Riemann Sum Types
Include Left, Right, Midpoint, and Trapezoidal methods based on how rectangle heights are chosen.
Calculation Steps
The process to compute a Riemann Sum, involving identifying the interval, calculating Δx, choosing a method, and evaluating the sum.
Practice with Different Sums
Engaging in exercises utilizing various Riemann Sum methods to gain mastery over the concepts.
Understanding Geometry
In the context of Riemann Sums, it refers to grasping how rectangles or trapezoids fit under a curve to estimate area.
Connection to Integrals
The link between Riemann Sums and definite integrals, transitioning from approximation to exact area.
Free-Response Questions
Open-ended questions on AP Calculus exams that require students to apply calculus concepts, including Riemann Sums.
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