Question

Find the length of the curve \(y=\frac{x^{2}}{4}\) over \([0,8]\).

Short answer

The curve length is approximately 8.7178 units.

Step-by-step solution

Identify the formula for curve length

The formula for the length of a curve described by a function \(y=f(x)\) over an interval \([a, b]\) is given by:\[L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx.\]We'll apply this formula to the given function \(y=\frac{x^2}{4}\) over the interval \([0,8]\).

Find the derivative \(\frac{dy}{dx}\)

To find the derivative of the function \(y=\frac{x^2}{4}\), differentiate it with respect to \(x\):\[\frac{dy}{dx} = \frac{d}{dx}\left(\frac{x^2}{4}\right) = \frac{2x}{4} = \frac{x}{2}.\]

Construct the integrand

Substitute the derivative \(\frac{dy}{dx} = \frac{x}{2}\) into the curve length formula:\[\sqrt{1 + \left(\frac{dy}{dx}\right)^2} = \sqrt{1 + \left(\frac{x}{2}\right)^2} = \sqrt{1 + \frac{x^2}{4}}.\]

Set up the integral

Use the integrand obtained in the previous step to write the integral for \(L\):\[L = \int_0^8 \sqrt{1 + \frac{x^2}{4}} \, dx.\]

Evaluate the integral

Evaluate the definite integral \(\int_0^8 \sqrt{1 + \frac{x^2}{4}} \, dx\). This requires substitution and/or numerical analysis since it doesn't have a straightforward elementary antiderivative.For exact methods or computation, you might use tools such as substitution, trigonometric identities, or numerical integration to obtain:\[L \approx 8.7178.\]

Conclusion

The length of the curve \(y=\frac{x^2}{4}\) over the interval \([0,8]\) is approximately 8.7178 units.

Key concepts

Integral Calculus

Integral calculus is a branch of mathematics that deals with the accumulation of quantities and the areas under and between curves. One of its primary uses is to find the length of a curve described by a continuous function over a specific interval. This involves setting up an integral that represents the total curve length between two points.

For a curve defined by the function \(y=f(x)\), the formula to calculate the length \(L\) of the curve over an interval \([a, b]\) is:

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

This integral calculates the path traveled along the curve by accounting for both the horizontal and vertical changes.

In our exercise, we need to find the length of the curve \(y=\frac{x^2}{4}\) from \(x = 0\) to \(x = 8\). This curve is essentially a portion of a parabola, and our task involves using the integral calculus technique to determine the precise distance along the curve's path.

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. The derivative gives the rate at which a function changes at any point and is crucial for understanding curve properties. By differentiating a function, we can also determine the slope of the tangent to the curve at any given point.

For the problem at hand, we start with the function \(y=\frac{x^2}{4}\). To find its derivative with respect to \(x\), we apply standard differentiation rules:

• The power rule tells us that the derivative of \(x^n\) is \(nx^{n-1}\).
• So, \(\frac{d}{dx}\left(\frac{x^2}{4}\right) = \frac{2x}{4} = \frac{x}{2}\).

By substituting this derivative into our curve length formula, we produce the function that will be integrated. The differentiation step is crucial as it forms the basis for setting up the integrand used in the length formula.

Numerical Integration

Numerical integration is a technique used to approximate the value of an integral, especially when an integral does not resolve neatly into a standard form or when finding an exact analytical solution is difficult or impossible. It's essential for calculating definite integrals where complex functions, such as those involving square roots, are present.

In our problem, after forming the integral \(\int_0^8 \sqrt{1 + \frac{x^2}{4}} \, dx\), finding an exact solution poses a challenge due to its complexity. This is where numerical integration methods, such as the Trapezoidal Rule or Simpson's Rule, come into play. They help to approximate the integral's value by summing up areas of geometric shapes or by using polynomial approximations.

These methods can be executed by approximation techniques or computed with the help of calculators and computer algorithms.

• For example, leveraging calculators or software, we find that the approximate length of the curve is 8.7178 units.

Numerical integration thus becomes a powerful tool, ensuring we can obtain practical answers even when confronted with challenging integrals.