Question

Set up the triple integral of the function over the solid inside the part of the sphere where . Make sure that you simplify the integrand.

a) ∭ f(x, y, z) dV
b) ∬ f(x, y) dA
c) ∫ f(t) dt
d) ∮ F • dr

Answer 1

The question pertains to setting up a triple integral over a three-dimensional space, with an emphasis on simplifying the integrand and understanding charge density in the context of electric fields.

Step-by-step explanation:

The question involves setting up a triple integral of a function over a three-dimensional solid, typically within a sphere in this context. The student is asked to simplify the integrand before performing the triple integral. For a particle in two dimensions, we use a double integral, but in three dimensions, we must use a triple integral to account for the volume.

When dealing with charge density that is not constant, it is essential to integrate over the volume enclosed by the Gaussian surface. For a spherical shell between radii r' and r' + dr', we consider the infinitesimal volume 4πr'^2 dr', and multiply it by the charge density at that point to find the charge within that shell. The correct expression for charge density may vary depending on location and must be appropriately applied in the integral.

The integrals assume the principle of superposition, which is crucial in electricity and magnetism calculations. Additionally, the dimension of an integral is the product of the dimension of the integrand and the dimension of the differential—this elucidates the units resulting from the integration process.