Line Integral Calculator

Visualize and calculate line integrals effortlessly. Enter your function, curve, and bounds to explore line integrals graphically and numerically.

Function (e.g., x + y)

Function to integrate. Use x and y as variables.

Curve Parameterization (e.g., [cos(t), sin(t)])

Parametric form of the curve. Use 't' as the parameter.

Lower Bound (e.g., 0)

Lower limit of the parameter t.

Upper Bound (e.g., 2*pi)

Upper limit of the parameter t.

Line Integral Value:

Visualization

Understanding Line Integrals

A line integral is an integral where the function to be integrated is evaluated along a curve. In simpler terms, instead of integrating over an interval on the x-axis, we integrate along a path. This tool calculates the line integral of a scalar function over a curve in 2D space.

Formula

For a scalar function $ f(x, y) $ and a curve $ C $ parameterized by $ \mathbf{r}(t) = [x(t), y(t)] $ for $ a \leq t \leq b $, the line integral is given by:

$$ \int_C f(x, y) \, ds = \int_a^b f(x(t), y(t)) \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt $$

How to Use This Calculator

Function: Enter the function $ f(x, y) $ you want to integrate. Use 'x' and 'y' as variables. Example: x^2 + y^2.
Curve Parameterization: Define the curve $ \mathbf{r}(t) = [x(t), y(t)] $ as a function of parameter 't'. Example: [cos(t), sin(t)] for a unit circle.
Lower and Upper Bounds: Specify the range of the parameter 't' over which to integrate. Example: 0 and 2*pi for a full circle.
Click 'Calculate' to compute the line integral. The result and a 3D visualization of the curve and function will be displayed.
Use 'Reset' to clear inputs and start a new calculation.
'Copy Result' to easily copy the calculated value.