Parametrizing the Intersection of Surfaces

Parametrize the curve of intersection of surfaces $Z=X^3$ and $Y=X^2$ from the origin to the point $(2,4,8)$ using $X=T$, $Y=T^2$, $Z=T^3$ for $T\in [0,2]$.

In this problem, we explore the intriguing concept of parametrizing curves, specifically focusing on the curve of intersection between two surfaces. Here, the surfaces given are $Z = X^3$ and $Y = X^2$. Parametrization is a method of introducing a parameter, usually denoted by $T$, to express the coordinates of each point on the curve uniquely. This allows us to traverse the curve smoothly and analytically. In this specific problem, the parameter $T$ is directly linked to the variable $X$, with $Y$ and $Z$ being dependent on $T$ as well. With $T$ taking values from $0$ to $2$, the parametrization method allows for a straightforward determination of coordinates, thus enabling a comprehensive examination from the origin to the point $(2, 4, 8)$.

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