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Chapter 5 | Integration
CHAPTER 5 REVIEW
KEY TERMS
average value of a function
(or f ave ) the average value of a function on an interval can be found by calculating the
definite integral of the function and dividing that value by the length of the interval
change of variables definite integral
the substitution of a variable, such as u , for an expression in the integrand a primary operation of calculus; the area between the curve and the x -axis over a given interval is a
definite integral
fundamental theorem of calculus
the theorem, central to the entire development of calculus, that establishes the
relationship between differentiation and integration
fundamental theorem of calculus, part 1 fundamental theorem of calculus, part 2
uses a definite integral to define an antiderivative of a function (also, evaluation theorem ) we can evaluate a definite integral by
evaluating the antiderivative of the integrand at the endpoints of the interval and subtracting
integrable function integrand integration by substitution lower sum mean value theorem for integrals Riemann sums as n goes to infinity exists the function to the right of the integration symbol; the integrand includes the function being integrated a technique for integration that allows integration of functions that are the result of a chain-rule derivative an approximation of the area under a curve computed by using the left endpoint of each subinterval to calculate the height of the vertical sides of each rectangle these values appear near the top and bottom of the integral sign and define the interval over which the function should be integrated a sum obtained by using the minimum value of f ( x ) on each subinterval guarantees that a point c exists such that f ( c ) is equal to the average value of the function if we know the rate of change of a quantity, the net change theorem says the future quantity is equal to the initial quantity plus the integral of the rate of change of the quantity the area between a function and the x -axis such that the area below the x -axis is subtracted from the area above the x -axis; the result is the same as the definite integral of the function a set of points that divides an interval into subintervals left-endpoint approximation limits of integration net change theorem net signed area a function is integrable if the limit defining the integral exists; in other words, if the limit of the
partition regular partition riemann sum
a partition in which the subintervals all have the same width an estimate of the area under the curve of the form A ≈ ∑ i =1 n
f ( x i * )Δ x
right-endpoint approximation
the right-endpoint approximation is an approximation of the area of the rectangles
sigma notation total area upper sum variable of integration index above and below the sigma indicate where to begin the summation and where to end it total area between a function and the x -axis is calculated by adding the area above the x -axis and the area below the x -axis; the result is the same as the definite integral of the absolute value of the function a sum obtained by using the maximum value of f ( x ) on each subinterval indicates which variable you are integrating with respect to; if it is x , then the function in the integrand is followed by dx (also, summation notation ) the Greek letter sigma (Σ) indicates addition of the values; the values of the under a curve using the right endpoint of each subinterval to construct the vertical sides of each rectangle
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