Question

A metallic surface s is in the shape of a hemisphere z = pr2 − x2 − y2, where (x, y) satisfies x2 y2 ≤ r2. the mass density (mass per unit area) at (x, y, z) ∈ s is given by m(x, y, z) = x2 y2. find the total mass of s

Answer 1

To calculate the total mass of the hemisphere, we integrate the given mass density function over the surface area described by the hemisphere's equation using either spherical or polar coordinates, depending on which simplifies the calculation.

Step-by-step explanation:

The student is asking about calculating the total mass of a hemispherical metallic surface described by a certain function, taking into account the mass density given by another function. To find the total mass of the hemisphere, we integrate the mass density function over the hemisphere's surface area.

The surface of the hemisphere is described by z = pr2 − x2 − y2 within the circular region x2 + y2 ≤ r2, and the mass density at any point (x, y, z) on this surface is given by m(x, y, z) = x2y2. We must set up a double integral over the circular region projected on the xy-plane to calculate the surface integral of this density function. This computes the total mass by summing the infinitesimal mass elements over the entire surface.

Since we're dealing with a hemisphere, the surface integral will typically involve spherical coordinates or polar coordinates for simplification. The bounds of integration would be the radii and angles that describe the full hemisphere.