lecture

Setup for Lecture 24 : Integration and Differentiation

One common way of talking about integration is to say we are
"solving for the area" bounded by one or more functions.

If we plot the two functions below, what is the "area" bounded by the two functions?

Often we are most interested in the area bounded by the two functions extends from the two values of x where the two functions cross each other. What do we call those two places?

How do we solve for the values of x for those two places using a solve block?

First set f(x) = g(x) using the boolean equal sign, then select solve from
the symbolic menu. What else do we need to type in to solve for x?

Once we know the roots of the two equations we can use the integration tools in MathCad
to solve for the area bounded by the two functions.

To solve for the area bounded by the two functions between x = -2 and x = 3
we use the "definite integral" symbol from the calculus toolbar.

When you first create the definite integral symbol, you see four place you have to fill in values or some other type of information.

The first piece of information to add are the functions used in the equations.

When you put the functions into the equation you will subtract the one on
the bottom from the function on the top.

In our example above, which function is the "top" and which function is the bottom?

The next piece of information to add is the range of values for which you are solving.
This range is what makes the equation a "definite" integral.

Enter these values into the black boxes at the top and the bottom of the
s-shaped integration symbol.

Last, we enter the symbol that represents the range for which we are solving: usually x

If we want, we can also assign the results to a variable.

It is possible to solve for the integral of a more complex area bounded by more than
two functions. Take for example the following three functions.

What would the plot of all three functions look like over the range x = 0 to x = 6?

What would be the roots of the the functions? Again, we use solve blocks to find the three roots, but this time, can we use f1(x), f2(x), and f3(x) instead of the equations?

Now that we know the x values of the roots, how do we set up the integrals?

Differentiation is commonly thought of as calculating a rate of change at some value
for a given function.

When trying to imagine what we mean by "rate of change" it helps to think of functions
as lines on a graph.

At any given point on a line, what value tells us what is the rate of change?

In MathCad we can calculate first order and higher order differential equations easily using
symbols for derivatives from the calculus toolbar.

Note: the symbols for both integration and differentiation were created by Gottfired Liebniz
in the 17th Century. The symbol for integration is a stylized S representing a "sum", and the
d in the symbol for derivatives represents "difference".

Again, when using this symbol from the calculus toolbar we need to add in information

First we add in the function we would like to differentiate and then the symbol we are
solving for.

Solving for a higher order derivative is just as easy, except we use the "nth order" derivative symbol and we need to also say whether we want a 2nd, 3rd, 4th, or whatever derivative.

Given a first-order or higher derivative, it is possible to solve for a numerical or a symbolic
solution.

If you have particular value of x you want to solve for, you use the plain old = sign.

If you want a symbolic solution, however you must make sure that x has
been redefined as x := x and use the arrow symbol from the symbolic toolbar.

A shortcut for getting the arrow is to press the keys ctrl and period

Using the symbolic arrow when x has been assigned a value still results in a numerical solution. You must redefine x as x:= x to get a symbolic solution.