Calculator Inputs
Enter the integrand using variables like x, y, z, r, theta, rho, phi, u, and v. You may use pi, sin, cos, sqrt, log, and similar functions.
Example Data Table
These examples help verify the calculator and show common Jacobian substitutions.
| Mode | Integrand | Bounds | Jacobian | Expected result |
|---|---|---|---|---|
| Polar | x^2 + y^2 | 0 ≤ r ≤ 1, 0 ≤ θ ≤ 2π | r | π / 2 |
| Cylindrical | 1 | 0 ≤ r ≤ 2, 0 ≤ θ ≤ 2π, 0 ≤ z ≤ 3 | r | 12π |
| Spherical | 1 | 0 ≤ ρ ≤ 1, 0 ≤ φ ≤ π, 0 ≤ θ ≤ 2π | ρ² sin(φ) | 4π / 3 |
| Affine 2D | 1 | x = 2u + v, y = u + 3v, 0 ≤ u, v ≤ 1 | |6 - 1| = 5 | 5 |
Formula Used
∬R f(x, y) dA = ∬S f(x(u, v), y(u, v)) |∂(x, y) / ∂(u, v)| du dv
∭V f(x, y, z) dV = ∭W f(x(u, v, w), y(u, v, w), z(u, v, w)) |J| du dv dw
x = r cos(θ), y = r sin(θ), and |J| = r
x = r cos(θ), y = r sin(θ), z = z, and |J| = r
x = ρ sin(φ) cos(θ), y = ρ sin(φ) sin(θ), z = ρ cos(φ), and |J| = ρ² sin(φ)
x = au + bv + c, y = du + ev + f, and |J| = |ae - bd|
The calculator evaluates the transformed integrand numerically using nested Simpson’s Rule. It substitutes the mapped coordinates into your function and multiplies by the absolute Jacobian determinant before integration.
How to Use This Calculator
- Choose the coordinate mode that matches your change of variables.
- Type the integrand using Cartesian variables x, y, and z when needed.
- Enter transformed bounds for the selected coordinates or variables.
- Set angle units to radians or degrees before submitting.
- Adjust Simpson points if you want a finer numeric estimate.
- Click the calculate button to view the result, summary table, and Plotly graph above the form.
FAQs
1. What does the Jacobian do in an integral?
It corrects the local area or volume scaling created by a variable transformation. Without it, the transformed integral would measure the wrong geometric size.
2. Should I enter the integrand in old variables or new variables?
Enter it in Cartesian form when possible. The calculator automatically substitutes transformed coordinates such as x = r cos(θ) and y = r sin(θ).
3. Why is my result slightly approximate?
This page uses nested Simpson’s Rule for numeric integration. Increasing the integration points usually improves accuracy for smooth functions and reasonable bounds.
4. When should I use polar coordinates?
Use polar coordinates for circular regions, radial symmetry, or expressions involving x² + y². They often simplify both the bounds and the integrand.
5. When should I use cylindrical or spherical coordinates?
Cylindrical coordinates help with cylinders and rotational solids. Spherical coordinates help with spheres, cones, and problems depending on distance from the origin.
6. Why does the affine transform require a nonzero determinant?
A zero determinant means the mapping collapses area into a line or point. That makes the transformation non-invertible and invalid for a standard Jacobian substitution.
7. Can I type expressions like 2*pi or sqrt(3)?
Yes. The input parser accepts common math expressions including pi, powers, roots, logarithms, and trigonometric functions with standard parentheses.
8. What do the CSV and PDF downloads contain?
They export the current result summary shown on the page. The example table also has its own CSV button for reference records or classroom handouts.