Introduction to Line Integral Calculator:
Line Integral Calculator is a digital tool that helps you to compute line integrals over vector fields or scalar functions in the given plane or in space. Our tool helps you to calculate the quantities like work done by a force along a path, flux of a vector field on a curve, and circulation of a vector field around a closed curve.
Line integral along curve calculator is a helpful tool as it can evaluate line integrals of both type functions for vector field or scalar fields over the curve C and numerical results without manual computation especially for complex paths.
What is a Line Integral?
A line integral is defined as an integral that evaluates the given function path in a scalar field or a vector field along a specified curve or path in space. It finds the value of work, flux, or circulation along the path defined by the curve.
It is also known as a path integral. Line integrals are an important concept in various fields of science and engineering, including physics, engineering, and mathematics.
$$ Work \;=\; \int_C F . dr \;=\; \int_a^b F(x(t), y(t), z(t)) . r’ (t) dt $$
- F: the function of the line integral
- r(t): the parametric value in x, y or z.
- d(t): the variable of line integral
How to Evaluate a Line Integral?
For the evaluation of line integral, integration method is used but before the integration process you need to find the value of r(t) in two or three dimensions as per the given value function.
It seems the process of line integral is complicated but when you follow our step by step calculation process of line integral function it will no longer be difficult. Nonetheless, let's see the calculation process of line integral method in steps.
Step 1: Identify the given function F(x, y, z) and the points that gives the r(t) value.
Step 2: To get the parametric value from the function point, use the equation as,
$$ r(t) \;=\; x(t)\; i + y(t)\; j + z(t)\; k $$
Step 3: After finding this r(t) value, differentiate it with respect to “t” to get the value of r`(t).
Step 4: Take the dot product of F . r`(t) to get a function for integration.
Step 5: Put the value F . r`(t) in the formula of line integral and evaluate the integral.
Step 6: After evaluation of integral, apply the limit to get the solution of line integral function.
For those who prefer not to manually calculate, we offer the line integral calculator with steps to quickly calculate the line integrals over vector fields.
Practical Example of Line Integral:
The below practical example of line integral will give you a better clarity about the line integral calculation process in steps.
Example: Find the work done by the vector field,
$$ \rightarrow{F}(x,y,z) \;=\; x \hat{i} + 3xy \hat{j} - (x + z) \hat{k} $$
On a particle moving along the line segment that goes from (1, 4, 2) to (0, 5, 1).
Solution:
Identify the given function
$$ F(x, y, z) \;=\; x_i + 3xy\; j - (x + z)\; k $$
The formula of line integral,
$$ Work \;=\; \int_C F . dr \;=\; \int_a^b F(x(t), y(t), z(t)) . r’ (t) dt $$
For the value of r(t) = ?
$$ r(t) \;=\; x(t)\; i + y(t)\; j + z(t)\; k $$
$$ r(t) \;=\; 〈1, 4, 2〉+ 〈0, 5, 1〉- 〈1, 4, 2〉]t \;=\; 〈1 - t, 4 + t, 2 - t〉$$
Differentiate with respect to t.
$$ f’(t) \;=\; -\hat{i} + \hat{j} - \hat{k} $$
Take the dot product of F . r(t) such as,
$$ F . r’(t) \;=\; -x + 3xy + x + z $$
$$ =\; 3xy + z $$
$$ =\; 3(1 - t)(4 + t) + (2 - t) $$
$$ =\; -3t^2 - 10t + 14 $$
Evaluate the integral with respect to t and apply the limit to get the solution of the given line integral function,
$$ \int_0^1 (-3t^2 - 10t + 14)\; dt $$
$$ \int_0^1 (-3t^2 - 10t + 14)\; dt \;=\; [-t^3 - 5t^2 + 14t]_0^1 \;=\; 8 $$
How to Use Line Integral of Vector Field Calculator:
Line integral curve calculator has a user-friendly interface that makes it easy for you for evaluating the line integral problems. Follow some instructions that are given as:
- Choose the variable of the given line integral function.
- Enter the function to find the solution of line integral in the input field.
- Recheck the given line integral problem before hitting the calculate button of line segment integral calculator to start the evaluation process.
- Click the “Calculate” button to get the result of your given line integral problem.
- If you are trying our line integral online calculator for the first time then you can use the load example to learn more about this method.
- Click on the “Recalculate” button to get a new page for finding more solutions of line integral problems.
Output of Line Integral Along Curve Calculator:
Line integral calculator with steps give you the solution from a given line integral problem when you add the input into it that may contain as:
- Result Option:
When you click on the result option it gives you a solution of the line integral problem.
- Possible Steps:
When you click on it, this option will provide you with the step by step solution of the line integral process.
Useful Features of Line Segment Integral Calculator:
Line integral curve calculator provides you with many useful features that help you to calculate the line integral problems. These features are:
- Line integral of vector field calculator is a free tool so you can use it for free to find the line integral problem without paying anything.
- It can manage various types of line integral function problems and calculate the solution.
- Our line integral calculator helps you to get conceptual clarity for the line integral process.
- It saves the time that you consume on the calculation of complex line integral problems.
- Line integral online calculator is a reliable tool that provides you with accurate solutions whenever you use it to calculate the line integral problem.
- It provides the solution without imposing any condition of signup after two to three usage.
