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Area Under a Curve definitions

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  • Area Under a Curve
    Region between a function's graph and the x-axis over a specified interval, representing accumulated value.
  • Linear Function
    Graph forms a straight line, allowing area calculations using basic geometric shapes like rectangles and triangles.
  • Interval
    Range on the x-axis, defined by starting and ending values, over which area is calculated.
  • Rectangle Approximation
    Method of estimating area under a curve by dividing the region into rectangles and summing their areas.
  • Left Endpoint Approximation
    Technique using the left side of each subinterval to determine rectangle heights for area estimation.
  • Right Endpoint Approximation
    Technique using the right side of each subinterval to determine rectangle heights for area estimation.
  • Delta x
    Width of each rectangle in an approximation, calculated as the interval length divided by the number of rectangles.
  • Summation
    Process of adding a sequence of terms, often used to combine areas of multiple rectangles.
  • Limit Definition
    Process of finding exact area by letting the number of rectangles approach infinity, eliminating approximation error.
  • Overapproximation
    Situation where the estimated area exceeds the true area, often due to rectangles extending above the curve.
  • Underapproximation
    Situation where the estimated area falls short of the true area, often due to rectangles missing parts of the curve.
  • Negative Area
    Result when a function lies below the x-axis over an interval, indicating area above the curve is counted as negative.
  • Base
    Horizontal length of a rectangle or geometric shape used in area calculations, typically corresponding to delta x.
  • Height
    Vertical measurement from the x-axis to the function value, determining the size of each rectangle in approximations.
  • x_k
    Specific x-value used to determine the height of the k-th rectangle in a sum, based on the chosen endpoint method.