partial integral calculator

What is a partial integral?

A partial integral is an integral of a multivariable function with respect to one chosen variable while treating the remaining variables as constants. For example, if you have f(x,y,z) and integrate with respect to x, then values of y and z are held fixed during the calculation.

How to use this calculator

1) Enter your function

Use standard math notation with variables x, y, and z. You may enter expressions like:

• x^2 + y*z
• sin(x) + exp(-x^2)
• sqrt(x + 1) + ln(y + 2)

2) Choose the integration variable

Select whether you want to integrate over x, y, or z. The calculator then sweeps that variable from lower bound a to upper bound b.

3) Set bounds and fixed variable values

Provide numerical bounds and optional values for the other variables. If a variable is not selected as the integration variable, it is treated as constant.

Numerical method used

This calculator uses Simpson's Rule, a reliable numerical integration method that approximates the area with piecewise quadratic curves. In practice, this gives high accuracy for smooth functions and converges quickly as interval count increases.

• Higher interval counts usually improve precision.
• An even interval count is required by Simpson's Rule (the tool auto-adjusts when needed).
• Functions with discontinuities or singularities inside bounds may produce errors or unstable values.

Common input mistakes

• Using invalid characters or unsupported symbols.
• Forgetting multiplication: write 2*x instead of 2x.
• Domain violations, such as sqrt(-1) in real numbers or ln(0).
• Choosing bounds that cross a singular point (for example, integrating 1/x through x=0).

Example interpretation

Suppose you evaluate ∫[0,2] (x^2 + y*z + sin(x)) dx with y=2 and z=3. Here, y*z=6 acts as a constant offset while only x changes across the interval. The resulting value combines polynomial growth, a constant baseline, and oscillation from the sine term.

When partial integrals are useful

• Physics: integrating density, flux, or energy density along one coordinate.
• Engineering: cross-sectional accumulation and parameter sweeps.
• Economics/data science: marginal accumulation when one variable changes and others stay fixed.

If you need full multivariable integration, compute nested partial integrals one variable at a time or switch to a multidimensional numerical method.