Calculated Results
These values appear above the form after submission, as requested.
| Metric | Value |
|---|
Sample Points Preview
| # | t | x(t) | y(t) | z(t) | dx/dt | dy/dt | dz/dt | Speed |
|---|
Calculator Inputs
Use explicit multiplication like 2*t. Supported functions include sin, cos, tan, sqrt, log, exp, abs, and pi.
Example Data Table
This example matches the default values loaded in the calculator.
| Curve | Integrand | Interval | Subintervals | ∫f(t)dt | ∫y dx | Arc Length | Enclosed Area |
|---|---|---|---|---|---|---|---|
| x(t)=3*cos(t), y(t)=3*sin(t), z(t)=0 | f(t)=t | 0 to 2*pi | 200 | 19.739209 | -28.274334 | 18.849556 | 28.274334 |
Formula Used
This calculator uses numerical differentiation and Simpson integration over the chosen parameter interval.
How to Use This Calculator
- Enter x(t), y(t), and z(t) if the curve has a third dimension.
- Type the general integrand f(t) for a standard parameter-based integral.
- Set the parameter start and end values using numbers or constants like pi.
- Choose an even Simpson subinterval count such as 200 or 400.
- Set preview rows and decimal places for reporting detail.
- Mark the curve as closed when enclosed area should be interpreted geometrically.
- Click Calculate Now to show results above the form.
- Use the export buttons to save the summary as CSV or PDF.
FAQs
What is a parametric integration calculator?
It evaluates quantities from x(t), y(t), and optional z(t) over a parameter interval. The tool converts curve-based expressions into numerical integrals with respect to t.
Which outputs does this calculator produce?
It computes a general integral, signed integrals with respect to x and y, arc length, enclosed area estimate, average speed, endpoint changes, and sample points.
Why can the signed area be negative?
The sign depends on traversal direction. Clockwise and counterclockwise motion reverse the sign, even when the geometric area remains the same.
Does this work for 3D curves?
Yes. The z(t) field is optional and contributes to arc length, speed, displacement, and general integration. Area calculations remain based on x(t) and y(t).
What functions can I type into expressions?
Use t, numbers, operators, parentheses, and functions like sin, cos, tan, sqrt, log, exp, abs, and pi. Write multiplication explicitly, such as 2*t.
How many subintervals should I use?
Start with 200 or 400 for smooth curves. Increase the value when the curve oscillates sharply, has steep slopes, or when you need tighter numerical accuracy.
Is enclosed area valid for open curves?
It is most meaningful for closed curves. For open paths, the result is a Green-style area estimate along the traced segment and should be interpreted carefully.
Why might results differ from textbook answers?
This file uses Simpson integration and numerical derivatives. Small differences can appear from interval count, rounding choices, or expressions with rapid local variation.