Multivariable Integral Calculator

Unravel complex integrals with ease. Calculate definite and indefinite integrals for functions with multiple variables.

Function

f(x, y, ...)

Enter the function with variables like x, y, z. Use standard math syntax.

Variable of Integration

variable

Specify the variable with respect to which you want to integrate.

Integration Limits (Optional)

[lower, upper]

Enter integration limits as a JSON array [lower, upper] for definite integrals. Leave empty for indefinite integrals.

Result:

Error:

Understanding Multivariable Integrals

Multivariable integrals extend the concept of single-variable integrals to functions of multiple variables. They are used to calculate volumes, areas, and other quantities in higher dimensions.

Definite Integrals: When limits are provided, the calculator computes the definite integral over the specified range for the given variable. For example, integrating with respect to 'x' from [0, 1] calculates the area under the curve along the x-axis within these limits, considering other variables as constants.

Indefinite Integrals: Without limits, the calculator finds the indefinite integral, which is a function whose derivative with respect to the chosen variable is the original function. This result will include an arbitrary constant of integration, which is omitted in the output for simplicity.

Example: For the function f(x, y) = x*y², integrating with respect to x would treat y as a constant. A definite integral from x=0 to x=2 would give a numerical value, while an indefinite integral would result in a function of x and y.