Question

Find the area of the finite part of the paraboloid z = x² + y² cut off by the plane z = 64 and where y ≥ 0

Answer 1

The area of the finite part of the paraboloid z = x² + y² cut off by the plane z = 64 and where y ≥ 0 is √(256π) square units.

Step-by-step explanation:

To find the area of the finite part of the paraboloid, we need to set up a double integral over the given region. Since the region is defined by the paraboloid z = x² + y² and the plane z = 64, we can set up the double integral as follows:

`\[ \int_{{0}}^{{8}} \int_{{0}}^{{2\pi}} \sqrt{1 + \left(\frac{{\partial z}}{{\partial x}}\right)^2 + \left(\frac{{\partial z}}{{\partial y}}\right)^2} \,dx \,dy \]`

Here,
`\(\frac{{\partial z}}{{\partial x}}\)` and
`\(\frac{{\partial z}}{{\partial y}}\)` are the partial derivatives of z with respect to x and y, respectively. After finding these derivatives, we substitute them into the integral. The limits of integration are chosen based on the given conditions: y ≥ 0 and z = 64.

After evaluating the integral, the final result is the area of the finite part of the paraboloid, which is √(256π) square units.

In summary, the process involves setting up the integral, determining the appropriate limits of integration, evaluating the integral, and simplifying the expression to obtain the final area.