Key Terms

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

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 $f\left(x\right)$ on each subinterval

mean value theorem for integrals

guarantees that a point c exists such that $f\left(c\right)$ 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 $A\approx {\displaystyle \sum _{i=1}^{n}f}(x_i^{*})\text{Δ}x$

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 $f\left(x\right)$ 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