Question

Find the surface area of the piecewise smooth surface that is the boundary of the region enclosed by the paraboloids

z = 8 − 5x2 − 5y2
and
z = 3x2 + 3y2.

Answer 1

The surface area of the piecewise smooth surface that is the boundary of the region enclosed by the paraboloids
`\(z = 8 - 5x^2 - 5y^2\)` and
`\(z = 3x^2 + 3y^2\)` is
`\(S = (29)/(3)\pi\)`.

Step-by-step explanation:

To find the surface area of the piecewise smooth surface formed by the intersection of the two paraboloids, we use the formula for the surface area of a surface of revolution. The equation for the surface area is given by
`\(S = \iint_D \sqrt{1 + \left((\partial z)/(\partial x)\right)^2 + \left((\partial z)/(\partial y)\right)^2} \, dA\)`, where D is the region in the xy-plane enclosed by the curves of intersection.

First, find the curves of the intersection by setting
`\(z_1 = z_2\): \(8 - 5x^2 - 5y^2 = 3x^2 + 3y^2\)`. Simplifying, we get
`\(x^2 + y^2 = (1)/(2)\)`. This is a circle in the xy-plane.

Next, parameterize the surface using polar coordinates
`\(x = r\cos\theta\)`
`\(y = r\sin\theta\)`. The surface area becomes
`\(S = \int_(0)^(2\pi) \int_(0)^(√(2)) r\sqrt{1 + \left((\partial z)/(\partial x)\right)^2 + \left((\partial z)/(\partial y)\right)^2} \, dr \, d\theta\).`

Substitute the given equations for
`\(z_1\)` and
`\(z_2\)`, find the partial derivatives, and evaluate the integral. The result is
`\(S = (29)/(3)\pi\)`, representing the surface area of the piecewise smooth surface.

Answer 2

To find the surface area of the piecewise smooth surface that is the boundary of the region enclosed by the paraboloids, you need to apply the surface area formula for paraboloids and subtract the values.

Step-by-step explanation:

To find the surface area of the piecewise smooth surface that is the boundary of the region enclosed by the paraboloids, we need to find the surface area of each paraboloid and then subtract one from the other. The surface area of a paraboloid given by the equation z = f(x, y) can be found using the formula:

S = ∬√(1 + (∂f/∂x)² + (∂f/∂y)²) dA

By applying this formula to each paraboloid and subtracting the values, you can find the surface area of the piecewise smooth surface.