Find the perimeter of the loop of the curve ay^2=x^2(a-x)

To find the perimeter of the loop of the curve defined by the equation \( ay^2 = x^2(a - x) \), we first need to analyze the shape and characteristics of the curve. This equation describes a loop, and we can derive its perimeter by using calculus, specifically the arc length formula.

Understanding the Curve

The given equation can be rearranged to express \( y \) in terms of \( x \). This will help us visualize the loop and determine the limits for our calculations. Rearranging gives us:

\( y^2 = \frac{x^2(a - x)}{a} \)

Taking the square root, we find:

\( y = \pm \sqrt{\frac{x^2(a - x)}{a}} \)

This indicates that for each \( x \), there are two corresponding \( y \) values, which confirms the presence of a loop.

Finding the Points of Intersection

Next, we need to determine the points where the curve intersects the x-axis. This occurs when \( y = 0 \), leading to:

\( 0 = \frac{x^2(a - x)}{a} \)

From this equation, we can see that \( x = 0 \) or \( x = a \). Thus, the loop exists between these two points.

Calculating the Arc Length

The formula for the arc length \( L \) of a curve defined by \( y = f(x) \) from \( x = a \) to \( x = b \) is given by:

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

To apply this, we first need to compute \( \frac{dy}{dx} \). Using implicit differentiation on the original equation:

\( 2ay \frac{dy}{dx} = 2x(a - x) - x^2 \frac{d}{dx}(a - x) \)

After simplifying, we can find \( \frac{dy}{dx} \) and then substitute it back into the arc length formula.

Setting Up the Integral

For our specific case, we will calculate the arc length for the upper half of the loop (from \( x = 0 \) to \( x = a \)) and then double it to account for the full perimeter:

\( L = 2 \int_0^a \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \)

Evaluating the Integral

Once we have \( \frac{dy}{dx} \), we can evaluate the integral. This may involve some algebraic manipulation and possibly numerical methods if the integral does not yield a simple antiderivative.

Final Steps

After evaluating the integral, we will arrive at a numerical value for the perimeter of the loop. This process highlights the importance of calculus in determining lengths of curves that are not straightforward geometric shapes.

In summary, the perimeter of the loop of the curve \( ay^2 = x^2(a - x) \) can be found through careful analysis of the curve, determining the necessary derivatives, and applying the arc length formula. Each step builds upon the previous one, leading to a comprehensive understanding of the curve's geometry.