Triple Integrals in Cylindrical and Spherical Coordinates

Cylindrical coordinates

Spherical coordinates

Choosing the system

The Jacobian in triple form

The cylindrical triple integral's volume element dV equals which expression?

The ball of radius a in spherical coordinates has which limits?

The triple integral of 1 over the ball of radius a in spherical coordinates gives which volume?

A solid consisting of a cylinder with a paraboloid's cap is most simply integrated in which coordinates?

Using the wrong coordinate system for a triple integral typically results in what?

The spherical volume element has rho squared sin phi. As a number, how many factors? Answer with a number.

The cone phi = pi/6 has the ice cream cone's phi ranging from 0 to pi/6. As a fraction of pi, what is the upper limit? Answer like pi/6.

The cylinder x squared + y squared = 4 has cylindrical r from 0 to what number? Answer with a number.

The ice cream cone solid, a cone topped by a spherical cap, in spherical coordinates has phi ranging 0 to the cone's half-angle and rho ranging 0 to what?

Spherical coordinates win when the solid involves spheres or cones centered where? Answer with one word.