Cylindrical Coordinates¶
We begin by defining cylindrical coordinates in $\mathbb{R}^3$.
Video Lecture
The Cylindrical Transformation¶
As we saw in the video above, cylindrical coordinates are essentially just polar coordinates in the $xy$-plane and rectangular coordinates in the $z$-direction. The cylindrical transformation is thus
$$
T(r,\theta,z) = \begin{cases}
x = r\cos\theta, \\[0.5 ex]
y = r\sin\theta, \\[0.5 ex]
z = z.
\end{cases}
$$
Because of this structure, the volume of a small "cylindrical box" can be decomposed into the area of a small polar rectangle times a small change in $z$,
$$
\Delta V_{ijk} = \Delta A_{ij}(r,\theta)\,\Delta z_k = r_i^* \Delta r_i \Delta \theta_j \Delta z_k.
$$
The volume element is thus $dV = r\,dr\,d\theta\, dz$.
Video Lecture
1. Example¶
Sketch the solid whose volume is given by the integral, then evaluate the integral.
$$
\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \int_0^2 \int_0^{r^2} r\,dz\,dr\,d\theta
$$
Check your answer
$$4\pi$$
Video Solution
Error Alert: There is a mistake near the very end of this video. In one of the last steps I accidentally transcribe a $2$ as a $3$, for no apparent reason.
2. Example¶
Evaluate $\int\!\!\int\!\!\int_E x^2\, dV$, where $E$ is the solid that lies within the cylinder $x^2 + y^2 = 1$, above the plane $z = 0$, and below the cone $z^2 = 4x^2 + 4y^2$.
Check your answer
$$\frac{2\pi}{5}$$
Video Solution
3. Example¶
Find the mass of a ball $B$ given by $x^2 + y^2 + z^2 \leq \rho^2$ if the density at any point is proportional to its distance from the $z$-axis.
Check your answer
$$\frac{K\pi^2\rho^4}{4},$$ where $K>0$ is a constant of proportionality.
Video Solution
Discussion¶
Questions? You can ask them here.