Question

Are length of a parabola Find the length of the curve \(y=a x^{2}\) from \(x=0\) to \(x=10,\) where \(a > 0\) is a real number.

Short answer

Answer: The formula for the arc length of a curve is \(L=\int_a^b \sqrt{1+(\frac{dy}{dx})^2} dx\). To apply it to find the length of the curve \(y=ax^2\) from \(x=0\) to \(x=10\), we first need to find the derivative of the curve with respect to x, which is \(\frac{dy}{dx}=2ax\). Then, we set up the integral as \(L=\int_0^{10} \sqrt{1+(2ax)^2} dx\) and simplify it to \(L=\int_0^{10} \sqrt{1+4a^2x^2} dx\). Calculating the arc length precisely requires the use of special functions like elliptic functions. Alternatively, numerical integration methods can be employed to approximate the length for a given value of \(a\).

Step-by-step solution

Find the derivative of the curve

To find the arc length, we need the derivative of the curve with respect to x. Given \(y=ax^2\), we can find the derivative as follows: \(\frac{dy}{dx}= \frac{d}{dx}(ax^2)=2ax\)

Set-up the formula for arc length

The formula for the arc length of a curve is given by: \(L=\int_a^b \sqrt{1+(\frac{dy}{dx})^2} dx\) In our case, we have: \(L=\int_0^{10} \sqrt{1+(2ax)^2} dx\)

Simplify the integral

Now, let's simplify the expression under the square root: \(L=\int_0^{10} \sqrt{1+4a^2x^2} dx\)

Finding the arc length

Unfortunately, the integral in its current form has no elementary solution. It requires the use of special functions like elliptic functions. Therefore, the length of the curve cannot be expressed as a simple elementary function of \(a\). However, it still can be solved numerically for specific values of \(a\). If necessary, we can use numerical integration methods such as Simpson's rule or the trapezoidal rule to find an approximation of the length for a given value of \(a\).

Key concepts

Derivative of a Function

Understanding the derivative of a function is essential when dealing with problems related to arc length. Imagine a curve defined by an equation like the parabola \( y = ax^2 \). The derivative, \( \frac{dy}{dx} \), represents the slope of the tangent to the curve at any point \( x \). For the equation \( y = ax^2 \), by applying the power rule in differentiation, you find that the derivative is \( \frac{dy}{dx} = 2ax \). The derivative tells us how steep the curve is at any point, which is crucial for calculating the arc length as it influences the distance along the curve. The steeper the slope, the longer the potential path. Thus, understanding derivatives not only tells us how a function behaves but also lays the groundwork for more complex calculations.

Arc Length Formula

The arc length formula is a key concept when you need to determine the length of a curve between two points. For a function \( y = f(x) \), the arc length \( L \) from \( x = a \) to \( x = b \) is computed using:

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

This formula combines the concept of horizontal movement with the vertical change provided by the derivative. By integrating, we sum all the small segments along the curve which takes into account both the horizontal distance traveled and the "rise" the derivative describes. In the case of the parabola \( y = ax^2 \), this becomes \( L = \int_0^{10} \sqrt{1 + (2ax)^2} \, dx \). Although this integral is complex and doesn't yield a straightforward solution, it forms the basis of calculating arc length and can be solved using advanced methods.

Numerical Integration Methods

When dealing with integrals that don't have simple solutions, numerical integration methods come to our rescue. These methods help us approximate the integral, providing us with a way to estimate the arc length of complex curves.

• Simpson’s Rule: This method provides a way to approximate the integral by using parabolic arcs rather than straight lines to better estimate the curve within each interval.
• Trapezoidal Rule: This method estimates the area under the curve by dividing it into trapezoids, summing their areas, and giving a more straightforward approach to approximating the integral.

Both methods are particularly useful when an integral lacks an elementary antiderivative, like in the case of \( \int_0^{10} \sqrt{1 + 4a^2x^2} \, dx \). By inputting specific values for \( a \), these techniques enable us to calculate the length of the curve to a degree of accuracy that is often sufficient for practical purposes. The choice of method can depend on the desired accuracy and computational efficiency.