Calculus Volume 1

Key Terms

Key Terms

average value of a function
(or fave) 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
the substitution of a variable, such as u, for an expression in the integrand
definite integral
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
uses a definite integral to define an antiderivative of a function
fundamental theorem of calculus, part 2
(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
a function is integrable if the limit defining the integral exists; in other words, if the limit of the Riemann sums as n goes to infinity exists
integrand
the function to the right of the integration symbol; the integrand includes the function being integrated
integration by substitution
a technique for integration that allows integration of functions that are the result of a chain-rule derivative
left-endpoint approximation
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
limits of integration
these values appear near the top and bottom of the integral sign and define the interval over which the function should be integrated
lower sum
a sum obtained by using the minimum value of $f(x)f(x)$ on each subinterval
mean value theorem for integrals
guarantees that a point c exists such that $f(c)f(c)$ is equal to the average value of the function
net change theorem
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
net signed area
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
partition
a set of points that divides an interval into subintervals
regular partition
a partition in which the subintervals all have the same width
riemann sum
an estimate of the area under the curve of the form $A≈∑i=1nf(xi*)ΔxA≈∑i=1nf(xi*)Δx$
right-endpoint approximation
the right-endpoint approximation is an approximation of the area of the rectangles under a curve using the right endpoint of each subinterval to construct the vertical sides of each rectangle
sigma notation
(also, summation notation) the Greek letter sigma (Σ) indicates addition of the values; the values of the index above and below the sigma indicate where to begin the summation and where to end it
total area
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
upper sum
a sum obtained by using the maximum value of $f(x)f(x)$ on each subinterval
variable of integration
indicates which variable you are integrating with respect to; if it is x, then the function in the integrand is followed by dx
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