f(x, y, z), over a three-dimensional region Rin xyz-space (three-dimensional space) is
called a triple integral. The notation is as follows:
Rf(x,y,z)dV
To compute the iterated integral, we need to integrate with respect to zfirst, then
y,then x. And when we integrate with respect to one variable, all the other variables
are assumed to be constant.DIFFERENTIAL EQUATIONS
What are differentialand ordinary differential equations?
Logically, a differential equationis one that contains differentials of a function, with
these equations defining the relationship between a function and one or more deriva-
tives of that function. More specifically, differential equations involve dependent vari-
ables and their derivatives with respect to the independent variables. To solve such
equations means to find a continuous function of the independent variable that, along
with its derivatives, satisfies the equation. An ordinary differential equationis one
that involves only one independent variable.What do the orderand degreeof a differential equationmean?
The orderof a differential equation is simply the highest derivative that appears in the
equation. The degreeof a differential equation is the power of the highest derivative
term. (For more information about power, see “Math Basics.”)What are implicitand explicit differential equations?
As seen above, an ordinary differential equation is one involving x, y, y', y',and so on.
Now add the idea that the order of the highest derivative is n. Thus, if a differential
equation of order nhas the form F(x, y', y",... y(n)) 0, then it is called an implicit
differential equation. If it is of the form F(x, y', y",... y(n1)) y(n), it is called an
explicit differential equation.What are some first-order differential equations?
A first-order differential equation is one involving the unknown function y,its deriva-
tive y', and the variable x. As seen above, these types of equations are usually referred
to as explicit differential equations.
There are several types of first-order differential equations, including separable,
Bernoulli, linear, and homogeneous (for explanations of the last two, see below). A
232 first-order differential equation takes the form: