## Introduction to Line Integral Calculator

Line integral calculator with steps is an advanced integral tool that helps you **find the work done** along a curved line in a vector field in two or three dimensions. It evaluates the integral functions whose surface area is enclosed under the curve path in a fraction of a second.

Line integrals are mathematical calculations that measure the cumulative effect of a vector field along a specific path or curve. Our calculator also simplifies the process of performing line integrals by allowing users to input the vector field and the curve of integration.

Addionally, our calculator measure the cumulative effect of a vector field along a specific path or curve. Fruther, for computations involving double integrals over regions in the plane or in space, you can use our double integral area calculator for efficient and precise solutions. Our calculator also offers a user-friendly interface and advanced algorithms to handle a wide range of double integral computations.

## What is a Line Integral?

Line integral is the **integral function** where integrands are multivariable functions or vector fields that lie along a path or space on a curve. This integral method is used to find the area under a closed curve along the curve line.

You can use the line integral technique to solve different types of equations with integration rules.It's particularly useful for students, researchers, and professionals working in fields such as physics, engineering, and mathematics, where line integrals are commonly used to analyze and solve various problems involving vector fields.

Further, for computations involving improper integrals, such as integrating over unbounded intervals or functions with singularities, you can utilize our improper integral solver to obtain accurate solutions and explore further mathematical analyses.

## Formula of Line Integral of Vector Field Calculator

The line integral formula helps to calculate work done using the integration method. The **formula** used by the line integral calculator is,

$$ \int_{C} \vec{F} . d \vec{r} \;=\; \int_{a}^{b} \vec{F} ( \vec{r} (t)) . \vec{r} (t) dt $$

Where,

This equation represents the line integral of a vector field F along a curve C parameterized by the vector function r(t) where t varies from a to b. For computations involving definite integrals over a specified interval, you can use our integral calculator definite for efficient and precise solutions.

- F is the vector field
- r(t) is the parametric equation of the curve line
- r'(t) is the derivative of the position vector r(t) on a curve
- dt is the variable of the integral
- a and b are the given upper and lower limits

## Working of Line Integral Curve Calculator

The line integral of vector field calculator has a simple working method to calculate the line integral problem for scalar and vector fields. It has an advanced algorithm that allows it to calculate various types of equations for line integral in just a few seconds.

When you **enter the input function** in the line segment integral calculator, it identifies the given function f(x,y,z) and a curve C (line segments) over which integration takes place. Now write the parametric equation r(t) of the curve as x(t),y(t),z(t).

Then, the line integral online calculator takes the determinate of the parametric equation |r'(t)|. Now it **puts the value** in the line integral formula and solves it to get the required solution of curve line.

Additionally, for calculations involving multiple integrals over regions in three-dimensional space, you can utilize our 3 integral calculator for efficient and precise solutions. Our calculator also provides a user-friendly interface and advanced algorithms to handle a wide range of triple-integral computations

Let's take an example of line integral to see the working method of line integral calculator with steps.

## Solved Example of Line Integral:

The line integral curve calculator can determine the line integral for you easily but sometimes you have to do it manually. For that, we are giving you an example to let you know about each step of manual calculation,

### Example:

An object moves from (-1,1) to (2,4) with the path r(t) = (t, t^{2}), with the force F = ＜x sin y, y＞. Calculate the work done.

**Solution:**

The force in terms of t can be written as ＜t sin(t^{2}), t^{2}＞, and evaluate r’(t) = ＜1,2t＞and then the work is,

$$ \int_{-1}^{2} (t \; sin(t^2), t^2) . (1,2t) dt \;=\; \int_{-1}^{2} t\; sin(t^2) + 2t^3 dt \;=\; \frac{15}{2} + \frac{cos(1) - cos(4)}{2} $$

We can write,

$$ \int_{C} x\;sin\;y\;dx + \int_{C} y\;dy \;=\; \int_{-1}^{2} x\;sin(x^2) dx + \int_{1}^{4} y\;dy $$

$$ - \frac{cos(4)}{2} + \frac{cos(1)}{2} + \frac{16}{2} - \frac{1}{2} $$

Thus it is the final solution of the given function limit. Addionally, for computations involving areas under curves or enclosed regions, you can use our calculator area under curve for efficient and precise solutions.our tool is particularly useful for determining the area enclosed by the curve described by the path r(t)-(t,t), providing a convenient means to visualize and analyze the geometric properties of the region.

## How to Use the Line Integral Calculator with Steps?

The line integral along curve calculator has a user-friendly interface that enables you to calculate the line integral easily. You should **follow our instructions** before using it for calculations. These instructions are:

Our calculator displays each step of the computation, aiding in understanding the process.For more exploration of related concepts in calculus, you can use our Indefinite Integral Calculator to find antiderivatives and reinforce the connection between integration and differentiation.

- Enter the parametric equation for the line integral in the respective field.
- Enter the line segment value from one point to another point
- Add the integral variable with respect to calculating the integration
- Click on the calculate button to get the solution of line integrals
- Press the recalculate button for more evaluation of the line integral

## Results from Line Segment Integral Calculator

Line integral calculator gives you the **outcome** of an integral question after adding the input to it. You can get some extra options along with the solution of the given integral problem. It may contain as:

- Result option provides you with the solution to the given integral problem of curvilinearity.
- Possible steps section provides you solution for the line integral.
- Plot section sketches a graph according to the given value of the line integral.

## Advantages of Line Integral Online Calculator:

The line integral of vector field calculator has provided millions of **benefits** when you use our line integral tool to solve areas that are present along a curve path. These benefits are:

Our calculator also serves as a valuable educational tool for students learning about line integrals, providing practical examples and explanations to enhance understanding of the concept. For further exploration of related concepts in calculus, you can utilize our calculate antiderivative, which provides solutions to antiderivative problems, reinforcing the connection between integration and differentiation.

- Our calculator is a trustworthy tool that provides you with accurate solutions of line integral questions
- It keeps us away from doing complex calculations of line integral problems
- This line integral curve calculator saves time because when you do lengthy integral evaluations manually, it consumes most of your time.
- It gives you the result of the line integral instantly.
- Line integral calculator with steps is free of cost tool, you can use it any time to solve integrals
- It has a wide range as it can solve various types of functions like parametric, multivariable functions… etc.

## Applications of Line Integral:

Line integral has different **applications in scientific fields**. It is mostly used in physics, calculus, and other scientific applications.

Moreover, visit our website calculate integration online for being attached with the variety of calculus tools.

- Mass of a wire
- Center of mass and Moments of inertia of a wire
- Work done by a force on an object moving in a vector field
- Ampere's law
- Faraday's law of magnetic induction