# Calc II Coverage Material



## 5.4: Triple Integrals



- Soln: 36
  ![image-20221023221544404](/home/zack/.config/Typora/typora-user-images/image-20221023221544404.png)

- Soln: 162
  ![image-20221023221621547](/home/zack/.config/Typora/typora-user-images/image-20221023221621547.png)
- Soln: 1/12
  ![image-20221023221730845](/home/zack/.config/Typora/typora-user-images/image-20221023221730845.png)
- Soln: 4/3
  ![image-20221023221754730](/home/zack/.config/Typora/typora-user-images/image-20221023221754730.png)
- Soln:  $$\int_{x=0}^{x=1} \int_{y=0}^{y=x^2} \int_{z=0}^{z=y^2} x y z \,\, d z d y d x = \int_{y=0}^{y=1} \int_{z=0}^{z=y^2} \int_{x=\sqrt{y}}^{x=1} x y z \,\,d x d z d y $$
  ![image-20221023221825185](/home/zack/.config/Typora/typora-user-images/image-20221023221825185.png)
- Soln: $\iiint_E \sqrt{x^2+z^2} d V=\int_{x=-2}^{x=2} \int_{y=x^2}^{y=4} \int_{z=-\sqrt{y-x^2}}^{z=\sqrt{y-x^2}} \sqrt{x^2+z^2} d z d y d x= 128 \pi / 15$
  ![image-20221023222009959](/home/zack/.config/Typora/typora-user-images/image-20221023222009959.png)
- Soln: ${147/40 \over 1/6}$.
  ![image-20221023222120775](/home/zack/.config/Typora/typora-user-images/image-20221023222120775.png)



Check even exercises.



## Cylindrical and Spherical Coordinates.

![image-20221023222458907](/home/zack/.config/Typora/typora-user-images/image-20221023222458907.png)



- Soln: $\int_{\theta=0}^{\theta=\pi / 2} \int_{r=0}^{r=2} \int_{z=0}^{z=4}(z r \sin \theta) r \,\,d z d r d \theta = 64/3$
  ![image-20221023222518005](/home/zack/.config/Typora/typora-user-images/image-20221023222518005.png)
- Soln: $=\int_{\theta=0}^{\theta=\pi} \int_{r=0}^{r=2 \sin \theta} \int_{z=0}^{z=\sqrt{16-r^2}} f(r, \theta, z) r d z d r d \theta$
  ![image-20221023222619467](/home/zack/.config/Typora/typora-user-images/image-20221023222619467.png)
- Soln: $V=\int_{\theta=0}^{\theta=2 \pi} \int_{r=0}^{r=1} \int_{z=r}^{z=2-r^2} r d z d r d \theta = \int_{\theta=0}^{\theta=2 \pi} \int_{z=0}^{z=1} \int_{r=0}^{r=z} r d r d z d \theta+\int_{\theta=0}^{\theta=2 \pi} \int_{z=1}^{z=2} \int_{r=0}^{r=\sqrt{2-z}} r d r d z d \theta$
  ![image-20221023222702627](/home/zack/.config/Typora/typora-user-images/image-20221023222702627.png)

- Soln: $V = \int_{\theta=0}^{\theta=2 \pi} \int_{r=0}^{r=1} \int_{z=0}^{z=\sqrt{4-r^2}} r d z d r d \theta = \int_{\theta=0}^{\theta=2 \pi} \int_{z=\sqrt{3}}^{z=2} \int_{r=0}^{r=\sqrt{4-r^2}} r d r d z d \theta+\int_{\theta=0}^{\theta=2 \pi} \int_{z=0}^{z=\sqrt{3}} \int_{r=0}^{r=1} r d r d z d \theta$
  ![image-20221023222848651](/home/zack/.config/Typora/typora-user-images/image-20221023222848651.png)





Spherical



![image-20221023223002150](/home/zack/.config/Typora/typora-user-images/image-20221023223002150.png)

![image-20221023222952199](/home/zack/.config/Typora/typora-user-images/image-20221023222952199.png)

![image-20221023223033569](/home/zack/.config/Typora/typora-user-images/image-20221023223033569.png)

![image-20221023223406878](/home/zack/.config/Typora/typora-user-images/image-20221023223406878.png)



- Soln: $2\pi/3$
  ![image-20221023223052831](/home/zack/.config/Typora/typora-user-images/image-20221023223052831.png)
- Soln: $=V(E)=\int_{\theta=0}^{\theta=2 \pi} \int_{\phi=0}^{\varphi=\pi / 6} \int_{\rho=0}^{\rho=2} \rho^2 \sin \varphi d \rho d \varphi d \theta$
  ![image-20221023223108668](/home/zack/.config/Typora/typora-user-images/image-20221023223108668.png)
- Soln: $V(E)=\int_{\theta=0}^{\theta=2 \pi} \int_{\rho=0}^{\rho=\sqrt{2} / 2} \int_{\varphi=0}^{\varphi=\pi / 4} \rho^2 \sin \varphi d \varphi d \rho d \theta+\int_{\theta=0}^{\theta=2 \pi} \int_{\rho=\sqrt{2} / 2}^{\rho=1} \int_{\varphi=0}^{\varphi=\cos ^{-1} \rho} \rho^2 \sin \varphi d \varphi d \rho d \theta = \pi/8$
  ![image-20221023223133721](/home/zack/.config/Typora/typora-user-images/image-20221023223133721.png)

- Soln: $=\int_{\theta=-\pi / 2}^{\theta=\pi / 2} \int_{r=0}^{r=1} \int_{z=r^2}^{z=r} r(r \cos \theta)(r \sin \theta) z d z d r d \theta$
  ![image-20221023223203515](/home/zack/.config/Typora/typora-user-images/image-20221023223203515.png)
- Soln: $y=\int_{\varphi=0}^{\varphi=\pi / 4} \int_{\theta=0}^{\theta=\pi / 2} \int_{\rho=0}^{\rho=3 \sqrt{2}} \rho^4 \sin \varphi d \rho d \theta d \varphi$
  ![image-20221023223237614](/home/zack/.config/Typora/typora-user-images/image-20221023223237614.png)
- Soln: $=8 \int_{\theta=0}^{\theta=\pi / 2} \int_{\rho=0}^{\rho=1} \int_{\varphi=0}^{\varphi=\pi / 2} a b c \rho^2 \sin \theta d \varphi d \rho d \theta$
  ![image-20221023223312217](/home/zack/.config/Typora/typora-user-images/image-20221023223312217.png)
- Problem:
  ![image-20221023223435834](/home/zack/.config/Typora/typora-user-images/image-20221023223435834.png)
  - ![image-20221023223519413](/home/zack/.config/Typora/typora-user-images/image-20221023223519413.png)