Find the surface area of the parametric surface defined by x = u^2 , y = u^3 and z = u over the interval [0,1]
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To find the surface area of the parametric surface defined by x = u^2, y = u^3, and z = u over the interval [0,1], we use the formula:
Surface Area = ∫∫ ||∂(r/∂u) × ∂(r/∂v)|| dA
where r(u,v) = (u^2, u^3, u) is the parameterization of the surface.
First, let's find the partial derivatives:
∂(r/∂u) = (2u, 3u^2, 1)
∂(r/∂v) = (0, 0, 0)
Next, we take the cross product of these partial derivatives:
(∂(r/∂u) × ∂(r/∂v)) = (0, 0, 0)
The cross product is the zero vector because the partial derivatives ∂(r/∂u) and ∂(r/∂v) are parallel to each other. This means that the surface does not have a tangent plane at every point and is not smooth.
Therefore, the surface area of this parametric surface is undefined.
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