integral
Compute symbolic and numerical integrals to find antiderivatives or calculate areas under curves. Supports definite and indefinite integration with exact analytical or approximate numerical methods.
Instructions
Compute symbolic and numerical integrals (definite and indefinite).
Examples:
INDEFINITE INTEGRAL (antiderivative): expression="x^2", variable="x" Result: "x^3/3"
DEFINITE INTEGRAL (area): expression="x^2", variable="x", lower_bound=0, upper_bound=1 Result: 0.333
TRIGONOMETRIC: expression="sin(x)", variable="x", lower_bound=0, upper_bound=3.14159 Result: 2.0 (area under one period)
NUMERICAL METHOD (non-elementary): expression="exp(-x^2)", variable="x", lower_bound=0, upper_bound=1, method="numerical" Result: 0.746824 (Gaussian integral approximation)
SYMBOLIC ANTIDERIVATIVE: expression="1/x", variable="x" Result: "log(x)"
Input Schema
TableJSON Schema
| Name | Required | Description | Default |
|---|---|---|---|
| context | No | Optional annotation to label this calculation (e.g., 'Bond A PV', 'Q2 revenue'). Appears in results for easy identification. | |
| output_mode | No | Output format: full (default), compact, minimal, value, or final. See batch_execute tool for details. | full |
| expression | Yes | Mathematical expression to integrate (e.g., 'x^2', 'sin(x)') | |
| variable | Yes | Integration variable (e.g., 'x', 't') | |
| lower_bound | No | Lower bound for definite integral (omit for indefinite) | |
| upper_bound | No | Upper bound for definite integral (omit for indefinite) | |
| method | No | Integration method: symbolic=exact/analytical, numerical=approximate (requires bounds) | symbolic |
Implementation Reference
- src/vibe_math_mcp/tools/calculus.py:95-125 (registration)MCP tool registration decorator for the 'integral' tool, defining name, detailed description with usage examples, and tool annotations.@mcp.tool( name="integral", description="""Compute symbolic and numerical integrals (definite and indefinite). Examples: INDEFINITE INTEGRAL (antiderivative): expression="x^2", variable="x" Result: "x^3/3" DEFINITE INTEGRAL (area): expression="x^2", variable="x", lower_bound=0, upper_bound=1 Result: 0.333 TRIGONOMETRIC: expression="sin(x)", variable="x", lower_bound=0, upper_bound=3.14159 Result: 2.0 (area under one period) NUMERICAL METHOD (non-elementary): expression="exp(-x^2)", variable="x", lower_bound=0, upper_bound=1, method="numerical" Result: 0.746824 (Gaussian integral approximation) SYMBOLIC ANTIDERIVATIVE: expression="1/x", variable="x" Result: "log(x)" """, annotations=ToolAnnotations( title="Integral Calculator", readOnlyHint=True, idempotentHint=True, ), )
- Input schema using Pydantic Annotated Fields defining parameters for the integral tool: expression, variable, optional bounds for definite integrals, and method (symbolic or numerical).async def integral( expression: Annotated[ str, Field( description="Mathematical expression to integrate (e.g., 'x^2', 'sin(x)')", min_length=1 ), ], variable: Annotated[ str, Field(description="Integration variable (e.g., 'x', 't')", min_length=1) ], lower_bound: Annotated[ float | None, Field(description="Lower bound for definite integral (omit for indefinite)") ] = None, upper_bound: Annotated[ float | None, Field(description="Upper bound for definite integral (omit for indefinite)") ] = None, method: Annotated[ Literal["symbolic", "numerical"], Field( description="Integration method: symbolic=exact/analytical, numerical=approximate (requires bounds)" ), ] = "symbolic", ) -> str:
- Core execution logic of the integral tool: parses expression with SymPy, computes symbolic integrals (definite/indefinite) using integrate(), numerical using SciPy quad(), builds metadata, formats result, handles errors."""Compute integrals using SymPy (symbolic/exact) or SciPy (numerical/approximate). Supports indefinite (antiderivatives) and definite (area) integrals.""" try: expr = sympify(expression) var = Symbol(variable) is_definite = lower_bound is not None and upper_bound is not None if method == "symbolic": if is_definite: # Definite symbolic integral result = integrate(expr, (var, lower_bound, upper_bound)) numeric_value = float(N(result)) metadata = { "expression": expression, "variable": variable, "lower_bound": lower_bound, "upper_bound": upper_bound, "symbolic_result": str(result), "type": "definite", } return format_result(numeric_value, metadata) else: # Indefinite symbolic integral result = integrate(expr, var) antiderivative_str = str(result) metadata = { "expression": expression, "variable": variable, "type": "indefinite", } return format_result(antiderivative_str, metadata) elif method == "numerical": if not is_definite: raise ValueError("Numerical integration requires lower_bound and upper_bound") # Convert SymPy expression to numeric function func = lambdify(var, expr, "numpy") # Use SciPy's quad for numerical integration result, error = integrate_numeric.quad(func, lower_bound, upper_bound) metadata = { "expression": expression, "variable": variable, "lower_bound": lower_bound, "upper_bound": upper_bound, "error_estimate": float(error), "method": "numerical", "type": "definite", } return format_result(float(result), metadata) else: raise ValueError(f"Unknown method: {method}") except Exception as e: raise ValueError( f"Integration failed: {str(e)}. " f"Example: expression='x^2', variable='x', lower_bound=0, upper_bound=1" )