$$ \iiint f(x,y,z)\, \text dz\, \text dy\, \text dx = \int \text dz \left[\int \text dy \left( \int f(x,y,z)\, \text dx\right) \right] $$

• The limits can be numbers, constants or functions of other variables
• innermost integral can have limits involving $y$ and $z$
• middle integral $\rightarrow z$
• outer integral will always have limits which are constants

Steps to solving multiple integrals

1. ✏️ Draw a diagram
2. 🔍 Identify the elemental area/mass/volume
3. 🐍 Find the limits
4. 🔁 Decide in which order to integrate
5. 🕵️ Check the limits follow the rules above
6. ✅ Do the integral and check answer

Example

• Evaluate the integral ${\int\int}_R x\,\text dx \text dy$ where $R$ is the region bounded by the curves: $y=(x-2)^2+2$ and $y=3$

✏️ Diagram:

🔍 We can see the relevant area in orange

🐍 $x$ limits are from $1 \to 3$ and $y$ limits are from $(x-2)^2+2\to3$

🔁 Here it obvious to integrate first by $y$ then $x$

🕵️ The limits follow the rules

✅ :

$$ \begin{aligned} &=\int_1^3 \text dx \left(\int^3_{(x-2)^2+2}x \,\text dy \right) \\ &=\int^3_1 -x^3+4x^2-3x\,\text dx \\ &=\frac{8}{3} \end{aligned} $$

Multi-variable integration using polar coordinates

<aside> <img src="https://em-content.zobj.net/source/microsoft-teams/337/hollow-red-circle_2b55.png" alt="https://em-content.zobj.net/source/microsoft-teams/337/hollow-red-circle_2b55.png" width="40px" /> Plane polar coordinates

</aside>

the values are represented by the respective colored lines

the values are represented by the respective colored lines

$$ \begin{aligned}x &= r\cos\theta \\y &= r\sin\theta \\\text dA &= r\, \text dr \,\text d\theta\end{aligned} $$

<aside> <img src="https://em-content.zobj.net/source/microsoft-teams/337/test-tube_1f9ea.png" alt="https://em-content.zobj.net/source/microsoft-teams/337/test-tube_1f9ea.png" width="40px" /> Cylindrical coordinates

</aside>

$$ \begin{aligned}x &= r\cos\theta \\y &= r\sin\theta \\ z &= z \end{aligned} $$

For curved surface: $\text dA=r \, \text d \theta \, \text dz$

For top surface: $\text dA=r \, \text d \theta \, \text dr$

For volume: $\text dV=r \, \text dr \, \text d \theta \, \text dz$

<aside> <img src="https://em-content.zobj.net/source/microsoft-teams/337/falafel_1f9c6.png" alt="https://em-content.zobj.net/source/microsoft-teams/337/falafel_1f9c6.png" width="40px" /> Spherical polar coordinates

</aside>

https://www.math3d.org/VqP8XpUQp

$$ \begin{aligned}x &= r\sin\theta\cos\phi \\y &= r\sin\theta\sin\phi \\z &= r\cos\theta \\ \text dA&=r^2\sin\theta\, \text d\theta\, \text d\phi \\ \text dV&=r^2\sin\theta\, \text dr\, \text d\theta \,\text d\phi \end{aligned} $$

Spherical polar coordinates

Example: Volume of a sphere

$$ \int^{2\pi}_0 \text d\phi \int^pi_0 \sin \theta \, \text d\theta \int^R_0 r^2\, \text dr = \frac{4}{3}\pi R^3 $$