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Length of Curve Calculator

Choose the type of formula and enter the values accordingly into the calculator to measure the length of a curve.

Choose type

Vector-valued f(x):

Lower Limit:

Upper Limit:

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Length of Curve Calculator

The Length of Curve Calculator finds the arc length of a curve over a given interval. It works with various types of curves, including Explicit, Parameterized, Polar, and Vector curves.

What is the Length of a Curve?

The length of a curve represents the total distance covered by an object moving from one point to another over a time interval [a, b]. It is also referred to as the arc length of a function.

Example: Consider the function y = f(x) = x² over the interval [2, 4].

Upper limit: 4
Lower limit: 2

The general formula for an explicit curve is:

$$ L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx $$

How to Find the Length of a Curve

Find the derivative of the function.
Compute the integral over the interval [a, b].

Explicit Curve: y = f(x)

For a function y = f(x) from x = a to x = b:

$$ \text{Arc length} = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx $$

Parameterized Curve

If the curve is given as x = f(t) and y = g(t) over t = a to t = b:

$$ \text{Arc length} = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt $$

Polar Curve

For a polar function r = r(t) with t ∈ [a, b]:

$$ L = \int_a^b \sqrt{r(t)^2 + \left(r'(t)\right)^2} \, dt $$

Vector Curve

For a 3D vector curve x = x(t), y = y(t), z = z(t) over t ∈ [a, b]:

$$ L = \int_a^b \sqrt{\left(x'(t)\right)^2 + \left(y'(t)\right)^2 + \left(z'(t)\right)^2} \, dt $$

Example

Find the length of the vector curve:

x = 17t³ + 15t² - 13t + 10
y = 19t³ + 2t² - 9t + 11
z = 6t³ + 7t² - 7t + 10
Interval: t ∈ [2, 5]

Step 1: Compute derivatives:

x'(t) = 51t² + 30t - 13
y'(t) = 57t² + 4t - 9
z'(t) = 18t² + 14t - 7

Step 2: Set up the integral:

$$ L = \int_2^5 \sqrt{(51t^2 + 30t - 13)^2 + (57t^2 + 4t - 9)^2 + (18t^2 + 14t - 7)^2} \, dt $$

How the Calculator Works

Input:

Choose the type of curve (Explicit, Parameterized, Polar, Vector)
Enter the function(s)
Enter the upper and lower limits
Click the Calculate button

Output:

Exact length of the curve over the specified interval

References

Wikipedia: Polar coordinate system
Tutorial Math Lamar: Arc Length Formulas

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