Exercise · ZW-CYLINDRICAL-AND-SPHERICAL-COORDINATES-010 · ★★★ Advanced
In which of these solids is it most convenient to use spherical coordinates ?
Choose one option.
A The cylinder \(x^2+y^2\le1\), \(0\le z\le5\) B The box \([0,1]^3\) C The solid under the plane \(x+y+z=1\) in the first octant D The region inside \(x^2+y^2+z^2\le4\) and outside the cone \(z=\sqrt{x^2+y^2}\) Show answer D. The region inside \(x^2+y^2+z^2\le4\) and outside the cone \(z=\sqrt{x^2+y^2}\)← Back to the question
Explanation The sphere and the cone become \(\rho=2\) and \(\varphi=\pi/4\), with constant limits . The cylinder calls for cylindrical coordinates , and the box and the tetrahedron are better described in rectangular ones.
Cylindrical coordinates \((r,\theta,z)\) and spherical coordinates \((\rho,\theta,\varphi)\) locate a point in space with distances and angles, and simplify solids with rotational symmetry.
\[\begin{aligned}x&=r\cos\theta,\ y=r\sin\theta,\ z=z\\x&=\rho\sin\varphi\cos\theta,\ y=\rho\sin\varphi\sin\theta,\ z=\rho\cos\varphi\end{aligned}\]
More practice problems: cylindrical and spherical coordinates ZW-CYLINDRICAL-AND-SPHERICAL-COORDINATES-001 A point has cylindrical coordinates (r,θ,z)=(2,π/2,5). What are its rectangular coordinates?ZW-CYLINDRICAL-AND-SPHERICAL-COORDINATES-002 The rectangular point (1,1,3) has, with r>0 and 0≤θ<2π, cylindrical coordinates:ZW-CYLINDRICAL-AND-SPHERICAL-COORDINATES-003 A point has spherical coordinates (rho,θ,φ)=(2,π/2,π/2). What are its rectangular coordinates?ZW-CYLINDRICAL-AND-SPHERICAL-COORDINATES-004 The cone z=√x^2+y^2 is written in spherical coordinates as:ZW-CYLINDRICAL-AND-SPHERICAL-COORDINATES-005 The sphere x^2+y^2+z^2=9 is written in cylindrical coordinates as:ZW-CYLINDRICAL-AND-SPHERICAL-COORDINATES-006 In cylindrical coordinates, the volume element is dV=ZW-CYLINDRICAL-AND-SPHERICAL-COORDINATES-007 Compute the volume of a ball of radius 2 by integrating in spherical coordinates.ZW-CYLINDRICAL-AND-SPHERICAL-COORDINATES-008 Compute iiint_E z dV, where E is the cylinder r≤1, 0≤ z≤2.