Drew Youngren dcy2@columbia.edu
\[ \iint\limits_{\mathcal R} f\,dA =\int_a^b \int_{g(x)}^{h(x)} f(x,y)\,dy\,dx. \]
\[ \iiint\limits_{\mathcal E} f\,dV =\int_a^b \int_{g(x)}^{h(x)} \int_{j(x,y)}^{k(x,y)}f(x,y,z)\,dz\,dy\,dx. \]
Let $E = [-1,1]^3$. Which of the following triple integrals is $0$?
\[ \begin{align*}&\iiint\limits_E (1 - x)\,dV & \iiint\limits_E e^{-yz}\,dV \\ &\iiint\limits_E yz^2\,dV & \iiint\limits_E x^2\,dV \end{align*}\]
Demo. Use the density feature to visualize the different integrands above.
Let $\mathcal E$ be the solid region in the first octant of $\RR^3$ bound by the surfaces $x + z = 1$ and $y = x^2$. Write the triple integral \[\iiint\limits_{\mathcal E} f\,dV\] as an iterated integral at least 2 ways.
Solution.
\[ \int_{0}^{1} \int_{0}^{x^2}\int_{0}^{1-x} f(x,y,z)\,dz\,dy\,dx \] \[ \int_{0}^{1} \int_{\sqrt y}^{1}\int_{0}^{1-x} f(x,y,z)\,dz\,dx\,dy \]\[\begin{align*} 0 &\leq r \lt \infty \\ 0 &\leq \theta < 2\pi \end{align*}\]
Consider a region with bounded polar coordinates $r_1 \leq r \leq r_2$ and $\theta_1 \leq \theta \leq \theta_2$.
\[ \Delta A = \frac12 r_2^2 (\theta_2 - \theta_1) - \frac12 r_1^2 (\theta_2 - \theta_1) \]
\[ \Delta A = \frac{r_1 + r_2}{2} (r_2 - r_1) (\theta_2 - \theta_1) \]
\[ \Delta A = \bar r\,\Delta r\, \Delta \theta \]
\[ \longrightarrow \]
\[dA = r\,dr\,d\theta\]
If $\mathcal R \subset \RR^2$ can be described with $\theta_1 \leq \theta \leq \theta_2$ and $g(\theta) \leq r \leq h(\theta)$ in polar coordinates, then \[ \iint\limits_{\mathcal R} f(x,y)\,dA = \int_{\theta_1}^{\theta_2} \int_{g(\theta)}^{h(\theta)} f(r \cos \theta, r\sin \theta)\,r\,dr\,d\theta \]
Remember to convert all parts of the integral:
Solve the volume problem from Lec 14 using a polar integral.
Solution. \[ V = \int_0^{\pi/2}\int_0^2 (4 - r^2)\,r\,dr\,d\theta = 2\pi \]
\[\begin{align*} 0 &\leq r \lt \infty \\ 0 &\leq \theta < 2\pi \\ -\infty &< z < \infty \end{align*}\]
Consider a region with bounded cylindrical coordinates $r_1 \leq r \leq r_2$, $\theta_1 \leq \theta \leq \theta_2$, and $z_1 \leq z \leq z_2$.
\[ \Delta V = \frac12 (r_2^2 - r_1^2) (\theta_2 - \theta_1) (z_2 - z_1) \]
\[ \Delta V = \bar r\,\Delta r\, \Delta \theta\,\Delta z \]
\[ \longrightarrow \]
\[dV = r\,dz\,dr\,d\theta\]
\[\begin{align*} 0 &\leq \rho \lt \infty \\ 0 &\leq \theta < 2\pi \\ 0 &\leq \phi \leq \pi \end{align*}\]
Write the point $(-1, -1, 1)$ in cylindrical and spherical coordinates.
Consider a region with bounded spherical coordinates $\rho_1 \leq \rho \leq \rho_2$, $\theta_1 \leq \theta \leq \theta_2$, and $\phi_1 \leq \phi \leq \phi_2$.
\[ \Delta V \approx (\Delta \rho)(\rho\Delta\phi)(\rho \sin \phi \Delta \theta) \]
\[ \longrightarrow \]
\[dV = \rho^2 \sin \phi \,d\rho\,d\phi\,d\theta\]
Sketch the region of integration in the definite integral below and then evaluate it using polar coordinates. \[\int_0^1\int_{\sqrt{1-x^2}}^{\sqrt{4-x^2}} x\,dy\,dx + \int_1^2\int_{0}^{\sqrt{4-x^2}} x\,dy\,dx \]
Solution \[ \int_0^{\pi/2} \int_1^2 r^2\cos\theta\,dr\,d\theta = \frac73 \]
Find the total mass of a solid region that is a right cone with base radius $R$, height $h$, and constant density $\mu$.
Solution. \[ \int_0^{2\pi} \int_0^R \int_0^{h - \frac{h}{R}r} \mu r \,dz\,dr\,d\theta = 2 \pi \mu (hr^2/2 - h r^3/(3R))\Big\vert_0^R\] \[ = \frac13 \pi R^2 h \mu \]
Find the average $z$ coordinate of the "half-ball" \[ \begin{cases} x^2 + y^2 +z^2 \leq R^2 \\ z \geq 0 \end{cases}. \]
We define the average value of a function $f$ on a region $\mathcal E$ to be \[ f_{\text{avg}} = \frac{\iiint_{\mathcal E} f\,dV}{\iiint_{\mathcal E} \,dV} \]
Solution.
We know the total volume is $\frac23 \pi R^3$.
\[ \iiint\limits_{\mathcal E} z\,dV = \int_0^{2\pi}\int_0^{\pi/2} \int_0^R (\rho \cos \phi)\rho^2 \sin \phi \,d\rho\,d\phi\,d\theta\]
\[ = 2\pi \frac{R^4}{4} \int_0^{\pi/2}\sin\phi \cos\phi\,d\phi = \frac14 \pi R^4 \]
Dividing, we get $z_\text{avg} = \frac38 R$.