Partial Integral Calculator
A partial integral calculator helps compute integrals of functions with multiple variables by integrating with respect to one variable while treating others as constants. This tool is essential in multivariable calculus, physics, and engineering for solving problems involving volume, mass, and probability density functions. It simplifies complex integration processes and provides step-by-step solutions, making it valuable for students and professionals working with partial differential equations and multivariate analysis.
Calculator
Formula
The partial integral formula for double integration: ∬Rf(x,y)dA = ∫ab∫cdf(x,y) dx dy Where integration order can be changed using Fubini's theorem when function is continuous over rectangular region R = [a,b]×[c,d].
How to Use
1. Enter function using variables x and y
2. Select integration variables
3. Input limits of integration
4. Click Calculate
5. View result with steps
Note: Use ^ for exponents (x^2) and * for multiplication. The calculator supports basic operations and trigonometric functions.
Derivation Process
Partial integration extends single-variable calculus to multiple dimensions. The process involves:
1. Fixing one variable constant
2. Integrating with respect to the first variable
3. Substituting limits for first integration
4. Repeating process for subsequent variables
Developed from Riemann sums in multiple dimensions, it uses Fubini's theorem to convert multiple integrals into iterated single integrals when continuous over rectangular regions.
Example Table
| Function | Integral |
|---|---|
| ∫∫x+y dx dy | 0.5x²y + 0.5xy² + C |
| ∫∫sin(xy) dx dy | -cos(xy)/(y²) + C |