Find the whole area of a loop of curve xy^2+(x+a)^2 (x+2a) = 0

To find the area of the loop defined by the curve given by the equation \( xy^2 + (x + a)^2 (x + 2a) = 0 \), we first need to analyze the equation and understand its structure. This equation is a polynomial in terms of \( x \) and \( y \), and it can be rewritten to isolate \( y \) in terms of \( x \). Let's break this down step by step.

Understanding the Equation

The equation can be rearranged as follows:

• Set \( xy^2 = - (x + a)^2 (x + 2a) \).
• Notice that the right-hand side is always non-positive since it is a product of squares and a linear term.

This implies that \( y^2 \) must also be non-positive, which means \( y^2 = 0 \) for the equation to hold true. Therefore, \( y = 0 \) is the only solution for \( y \), leading us to the x-values where the curve intersects the x-axis.

Finding the x-Intercepts

To find the x-intercepts, we set \( y = 0 \) in the original equation:

Substituting \( y = 0 \) gives us:

\( (x + a)^2 (x + 2a) = 0 \).

This equation can be solved by setting each factor to zero:

• \( (x + a)^2 = 0 \) gives \( x = -a \) (with multiplicity 2).
• \( (x + 2a) = 0 \) gives \( x = -2a \).

Identifying the Loop

The points \( x = -a \) and \( x = -2a \) are critical for understanding the loop. The curve forms a loop between these two points on the x-axis. The area enclosed by the loop can be calculated using integration.

Calculating the Area

The area \( A \) can be found using the integral of the function that describes the curve. Since \( y = 0 \) for the loop, we need to consider the contributions from the x-values:

To find the area, we can use the formula:

\( A = \int_{-2a}^{-a} |y| \, dx \).

However, since \( y = 0 \), we need to consider the contributions from the curve itself. The area can be computed as:

\( A = \int_{-2a}^{-a} \sqrt{-\frac{(x + a)^2 (x + 2a)}{x}} \, dx \).

This integral will give us the area of the loop. To compute it, we can simplify the expression under the square root and evaluate the definite integral.

Final Steps

After performing the integration, you will arrive at a numerical value representing the area of the loop. The exact computation may involve substitution or numerical methods depending on the complexity of the integral.

In summary, the area of the loop defined by the curve \( xy^2 + (x + a)^2 (x + 2a) = 0 \) can be found by determining the x-intercepts, identifying the loop, and calculating the area using integration techniques. This approach not only provides the area but also deepens your understanding of how curves behave in the Cartesian plane.