Arc Length Calculator

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Example

Created on 2024-06-20Asked by Sofia Garcia (Solvelet student)

Find the arc length of the curve

y = \frac{1}{2}x^2

from

x = 0

x = 2

Solution

Step 1: Identify the function and the limits of integration. The function is

y = \frac{1}{2} x^2

and the limits are

x = 0

x = 2

. Step 2: Find the derivative of the function:

\frac{dy}{dx} = x.

Step 3: Set up the arc length integral:

\int_{0}^{2} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx = \int_{0}^{2} \sqrt{1 + x^2} \, dx.

Step 4: Evaluate the integral using the substitution

x = \sinh(t)

dx = \cosh(t) \, dt

\int_{0}^{2} \sqrt{1 + x^2} \, dx = \int_{0}^{\text{arsinh}(2)} \cosh^2(t) \, dt.

Step 5: Use the identity

\cosh^2(t) = \frac{1}{2} (\cosh(2t) + 1)

to integrate:

\int_{0}^{\text{arsinh}(2)} \frac{1}{2} (\cosh(2t) + 1) \, dt = \left. \frac{1}{4} \sinh(2t) + \frac{1}{2} t \right|_{0}^{\text{arsinh}(2)}.

Step 6: Apply the limits of integration:

\frac{1}{4} \sinh(2 \cdot \text{arsinh}(2)) + \frac{1}{2} \text{arsinh}(2) = \frac{1}{4} \cdot 4 \cdot \sqrt{5} + \frac{1}{2} \cdot \text{arsinh}(2) = \sqrt{5} + \frac{1}{2} \text{arsinh}(2).

Step 7: Conclusion. The arc length of the curve

y = \frac{1}{2} x^2

from

x = 0

x = 2

\sqrt{5} + \frac{1}{2} \text{arsinh}(2)

. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Lucas Allen on Solvelet

1. Find the curve length

y = x^{3//2}

from

x = 0

x = 4

2. Calculate the arc curve length defined by the parametric equations

x = \cos(t)

y = \sin(t)

over the interval

[0, \pi]

DefinitionPath Length- Length along a (curved) path in a plane is measured by this. It computes by using the Euclidean distance formula and integrating over the parameterization of the curves. Arc length is useful for evaluating the distance traveled in physics, engineering, or wherever you would like to measure path distance. Applications include finding the work done by a force, analysis of curves in space, etc. The above formula is used in calculus to find the arc length of a function \( y = f(x)\), \( a <= x <= b \) where \( a \) & \( b \) are the x-coordinates.

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