Exercise · ZW-CYLINDRICAL-AND-SPHERICAL-COORDINATES-009 · ★★★ Advanced
Compute \(\iiint_E z\,dV\), where \(E\) is the upper half-ball \(x^2+y^2+z^2\le1\), \(z\ge0\).
Choose one option.
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- B. \(\tfrac{\pi}{4}\)
Explanation
In spherical coordinates, \(\int_0^{2\pi}\!\int_0^{\pi/2}\!\int_0^1(\rho\cos\varphi)\rho^2\sin\varphi\,d\rho\,d\varphi\,d\theta=2\pi\cdot\tfrac14\cdot\tfrac12=\tfrac{\pi}{4}\). The integrand contributes \(\rho^3\), not \(\rho^2\).
Review the concept: Cylindrical and spherical coordinates
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.
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