### Key Terms

- average value of a function
- (or
) 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*f*_{ave}

- 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 $A\approx {\displaystyle \sum _{i=1}^{n}f}({x}_{i}^{*})\text{\Delta}x$

- 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*