Unit Six: Integration and Accumulation of Change- essential knowledge

What does the area of the region between the graph of a rate of change function and the x-axis represent?

The accumulation of change

How can accumulation of change sometimes be evaluated?

by using geometry.

What is the accumulated change if a rate of change is positive (negative) over an interval

The accumulated change is positive (negative).

What is the unit for the area of a region defined by a rate of change?

It is the unit for the rate of change multiplied by the unit for the independent variable.

How can definite integrals be approximated?

For functions that are represented graphically, numerically, analytically, and verbally.

Name four methods to approximate definite integrals.

Left Riemann sum, right Riemann sum, midpoint Riemann sum, or trapezoidal sum.

Can definite integrals be approximated using numerical methods?

Yes, with or without technology.

How can the behavior of a function affect the approximation of a definite integral?

It may be possible to determine whether an approximation is an underestimate or overestimate for the value of the definite integral.

How can the limit of an approximating Riemann sum be interpreted?

As a definite integral.

What is a Riemann sum?

It is the sum of products, each of which is the value of the function at a point in a subinterval multiplied by the length of that subinterval of the partition.

How is the definite integral of a continuous function fff over the interval [a,b][a, b][a,b] denoted?

As ∫abf(x) dx, which is the limit of Riemann sums as the widths of the subintervals approach 0.

How can a definite integral be translated and written?

It can be translated into the limit of a related Riemann sum, and the limit of a Riemann sum can be written as a definite integral.

What can the definite integral be used to define?

New functions.

If f is a continuous function on an interval containing aaa, what is d/dx[∫axf(t) dt]

f(x), where x is in the interval.

What types of representations of a function f provide information about the function g defined as g(x)=∫axf(t) dt

Graphical, numerical, analytical, and verbal representations.

How can a definite integral sometimes be evaluated?

By using geometry and the connection between the definite integral and area.

What are some properties of definite integrals?

The integral of a constant times a function, the integral of the sum of two functions, reversal of limits of integration, and the integral of a function over adjacent intervals.

To what types of functions can the definition of the definite integral be extended?

Functions with removable or jump discontinuities.

What is an antiderivative of a function f?

A function g whose derivative is f

If f is continuous on an interval containing aaa, what is F(x)=∫axf(t) dt

An antiderivative of f for x in the interval

If f is continuous on the interval [a,b] and F is an antiderivative of f, what is ∫abf(x) dx

F(b)−F(a)

How can an indefinite integral of the function f be expressed?

As ∫f(x) dx=F(x)+C, where F′(x)=f(x) and C is any constant

What provides the foundation for finding antiderivatives?

Differentiation rules.

Do all functions have closed-form antiderivatives?

No, many functions do not have closed-form antiderivatives.

What is a technique for finding antiderivatives?

Substitution of variables.

What does substitution of variables require for a definite integral?

Corresponding changes to the limits of integration.

What are some techniques for finding antiderivatives?

Rearrangements into equivalent forms, such as long division and completing the square.