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Parametric Arc Length Calculator + Online Solver With Free Steps
A Parametric Arc Length Calculator is used to calculate the length of an arc generated by a set of functions. This calculator is specifically used for parametric curves, and it works by getting two parametric equations as inputs.
The Parametric equations represent some real-world problems, and the Arc Length corresponds to a correlation between the two parametric functions. The calculator is very easy to use, with input boxes labeled accordingly.
What Is a Parametric Arc Length Calculator?
A Parametric Arc Length Calculator is an online calculator that provides the service of solving your parametric curve problems.
These parametric curve problems are required to have two parametric equations describing them. These Parametric Equations may involve x(t) and y(t) as their variable coordinates.
The Calculator is one of the advanced ones as it comes in very handy for solving technical calculus problems. There are input boxes given in this Calculator and you can enter your problem’s details in them.
How To Use a Parametric Arc Length Calculator?
To use a Parametric Arc Length Calculator, you must first have a problem statement with the required parametric equations and a range for the upper and lower bounds of integration. Following that, you can use the Parametric Arc Length Calculator to find your parametric curves’ Arc lengths by following the given steps:
Step 1
Enter the parametric equations in the input boxes labeled as x(t), and y(t).
Step 2
Next, enter the upper and lower limits of integration in the input boxes labeled as Lower Bound, and Upper Bound.
Step 3
Then, you can simply press the button labeled Submit, and this opens the result to your problem in a new window.
Step 4
Finally, if you would like to keep using this calculator, you can enter your problem statements in the new intractable window and get results.
How Does a Parametric Arc Length Calculator Work?
A Parametric Arc Length Calculator works by finding the derivatives of the parametric equations provided and then solving a definite integral of the derivatives correlation. After solving everything, the calculator provides us with the arc length of the Parametric Curve.
Parametric Curve
A Parametric Curve is not too different from a normal curve. The main difference between them is the representation. In a Parametric Curve, we use a different variable to express the correlation between its x and y coordinates.
Arc Length
Arc Length is a significant value in the fields of Physics, Mathematics, and Engineering. Using Arc Length, we can make certain predictions and calculate certain immeasurable values in real-life scenarios.
For example, finding out the trajectory of a rocket launched along a parabolic path is something only Arc Length can help us with, and keeping this Arc Length in a parametric form only helps with managing the variables under question.
The Arc Length solution to a problem of this sort: fx = x(t), fy = y(t) is given by the following expression:
\[L_{arc} = \int_{a}^{b} \sqrt {(\frac {dx(t)}{dt})^2 + (\frac {dy(t)}{dt})^2} \,dt\]
Solved Examples:
Example 1
Consider the given parametric equations:
\[x(t) = -\sqrt(t), y(t) = 1-t\]
And solve for Arc Length in the range 0 to 9.
Solution
Our curve is described by the above parametric equations for x(t) and y(t). To find the Arc Length, we must first find the integral of the derivative sum given below:
\[L_{arc} = \int_{a}^{b} \sqrt {(\frac {dx}{dt})^2 + (\frac {dy}{dt})^2} \,dt\]
Placing our values inside this equation gives us the arc length $L_{arc}$:
\[L_{arc} = \int_{0}^{9} \sqrt {\bigg(\frac {d(-\sqrt{t})}{dt}\bigg)^2 + \bigg(\frac {d(1-t)}{dt}\bigg)^2} \,dt = \int_{0}^{9}\sqrt{1 + \frac{1}{4t}} \,dt \approx 9.74709\]
Example 2
Consider the given parametric equations:
\[x(\theta) = 2 \cos^2 (\theta), y(\theta) = 2 \cos (\theta) \sin (\theta)\]
And solve for Arc Length in the range 0 to $\pi$.
Solution
The curve is described by the following parametric equations for x(t) and y(t), respectively:
\[x(\theta) = 2 \cos^2 (\theta)\]
\[ y(\theta) = 2 \cos (\theta) \sin (\theta)\]
To find the Arc Length, we must first find the integral of the derivative sum given below:
\[L_{arc} = \int_{a}^{b} \sqrt {(\frac {dx}{d\theta})^2 + (\frac {dy}{d\theta})^2} \,d\theta\]
Input the values inside this equation.
The arc length $L_{arc}$ is given as:
\[L_{arc} = \int_{0}^{\pi} \sqrt {\bigg(\frac {d(2 \cos^2 (\theta))}{d\theta}\bigg)^2 + \bigg(\frac {d(2 \cos (\theta) \sin (\theta))}{d\theta}\bigg)^2} \,d\theta = \int_{0}^{\pi}2 \,d\theta \approx 6.28\]