Symbolic
Integration
Static
symbolic evaluation - using the Symbolics menu
Live
Symbolic Evaluation - using the Symbolic toolbar
Difference
Between Static and Live Symbolic Evaluation
A
powerful feature of Mathcad is the ability to do symbolic algebra and calculus,
which means exact arithmetic. Scientist and engineers commonly want a numerical
value as the result of a calculation, and Mathcad numerical math features provide
this type of result. However there are times when you need a result in terms
of the mathematical symbols themselves, and you can use Mathcad's symbolic math
capability for that type of calculation.
Mathcad symbolic commands are located in the Symbolics menu and also in the Symbolic toolbar.
Symbolic calculations can either be static or live. A live symbolic calculation is automatically updated when a change is made to the document; a static calculation is not updated. Static symbolic work is accomplished by using the commands on the Symbolics menu; live symbolic calculation is accomplished using the symbolics equal sign (the right arrow), or other tools on the Symbolics palette.
Static
symbolic evaluation - using
the Symbolics menu
To evaluate an indefinite integral:
- Click the Indefinite Integral symbol on the Calculus toolbar.
- Fill in the placeholders for the integrand and the integration variable.
- Press Space until the entire integral is between the editing lines.
- Choose Evaluate/Symbolically from the Symbolics menu. Mathcad will display the result.
To evaluate a definite integral, click the Definite Integral symbol on the Calculus Toolbar and follow the instructions above. The symbolic processor will find an indefinite integral and subtract its value at the lower limit of integration from its value at the upper limit. If you use decimals in the integrand or limits of integration, you'll see a 20-place numerical approximation if one is available.
When evaluating an indefinite integral, remember that the answer to an integration problem is not unique. If f(x) is an integral of a given function, so is f(x) + C for a constant C.
Note:
When
you use symbolic commands from the Symbolics menu, the symbolic processor does
not "know" about any variable definitions in your worksheet. If you
need that kind of live symbolic evaluation, use live
symbolic evaluation
instead of the menu commands.
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Live
Symbolic Evaluation - using the Symbolic toolbar
To perform Live Symbolic Evaluation use the symbolic equal sign (the right arrow) in the Symbolic toolbar:
- Type the expression you want to evaluate symbolically.
- Click the symbolic equal sign (the right arrow) in the Calculus toolbar.
- Press Enter.
Difference
Between Static and Live Symbolic Evaluation
By default, the symbolic equal sign returns the same expression as that returned by choosing Evaluate Symbolically from the Symbolics menu, but it takes other definitions in your worksheet into account.
Let's define the following function:
and evaluate the following integral in two different ways:
a. Using the static symbolic evaluation:
Select the integral on the left, then select Evaluate / Symbolically from the Symbolics menu.
The symbolic transformation was unsuccessful, so the new expression is identical to the old one.
The
symbolic processor from the
Symbolics menu does not "know" about any variable or function definitions
in your worksheet, so it is not "aware" of
f(t) := ln(3t) function definition. It can not evaluate symbolically the integral above.
b. Using the live symbolic evaluation:
Select the integral below, then click the Symbolic Evaluation symbol (the right arrow) in the Symbolic toolbar.
Symbolic Evaluation from the Symbolic toolbar is "aware" of all the definitions in your worksheet, so it can evaluate the above integral symbolically.
Example 2
Static symbolic evaluation is not aware
of the a and b values and yields:
Live symbolic evaluation is aware of the a and b definitions and yields:
