91Ó°ÊÓ

When dealing with three-dimensional figures, understanding how they intersect is crucial. Here, we are examining the intersection between a paraboloid and a plane. The paraboloid is given by the equation \( x^2 + y^2 = z \), while the plane is represented by \( z = 2 \). Replacing \( z \) in the paraboloid equation with 2 gives us \( x^2 + y^2 = 2 \). This equation describes a circle with radius \( \sqrt{2} \). This circle represents the path along which the plane slices through the paraboloid. Grasping this intersection helps in visualizing the surface area trapped "above" this circle, as we further calculate.

Polar coordinates offer a simplified method to handle calculations involving circles and curves. Instead of relying on rectangular coordinates (\(x, y\)), polar coordinates use a distance \(r\) from the origin and an angle \(\theta\) from the polar axis. Here, we use polar coordinates to represent every point on the intersection circle \( x^2 + y^2 = 2 \). The switch to polar coordinates,

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)

makes it convenient to integrate circular regions. For this integration, \( r \) ranges from 0 to \( \sqrt{2} \), and \( \theta \) spans from \( 0 \) to \( 2\pi \). This setup aligns perfectly with the form of the paraboloid as we examine the surface contained within the bounding circle.

A parametric representation is a method of expressing a surface with parameters \( r \) and \( \theta \). This approach is particularly useful for integrating over surfaces in three dimensions. For the paraboloid \( z = x^2 + y^2 \), the parametric representation becomes \( \mathbf{r}(r, \theta) = (r \cos \theta, r \sin \theta, r^2) \). This formula maps the surface in terms of \( r \) and \( \theta \), helping us visualize and compute properties like surface area. As we change \( r \) and \( \theta \), this parametric form allows us to accurately describe the 3D surface, essentially "drawing" the paraboloid section using geometry and calculus.

Integral calculus is an essential tool in finding the area of complex surfaces. In this problem, we integrate over the parametric representation of the paraboloid described using polar coordinates. After setting up the bounds and the parametric form, the focus moves to finding the area by evaluating an integral. We compute the partial derivatives of the parametric surface, apply a cross product, and determine its magnitude. This magnitude, representing differential area elements, forms the integral's core component:

• Cross Product: Computes surface orientation
• Magnitude: Represents the differential area

The final integral, \[ A = \int_0^{2\pi} \int_0^{\sqrt{2}} r\sqrt{4r^2 + 1} \, dr \, d\theta \], evaluates the surface area by integrating first with respect to \( r \) and subsequently \( \theta \). This process simplifies the problem, turning a theoretical surface problem into computations using algebra and calculus principles.