From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Thu, 9 May 2019 22:51:04 +0000 (+1000)
Subject: [spec] solids of revolution
X-Git-Tag: yr12~144
X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/29ec3836a781a8ef83d7ec78803e2f0b08f5d037?ds=sidebyside

[spec] solids of revolution
---

diff --git a/spec/calculus.md b/spec/calculus.md
index 8b4fcb8..63b2e90 100644
--- a/spec/calculus.md
+++ b/spec/calculus.md
@@ -322,6 +322,29 @@ Restrictions: `Define f(x)=...` $\rightarrow$ `f(x)|x>1` (e.g.)
 
 To find stationary points of a function, substitute $x$ value of given point into derivative. Solve for ${dy \over dx}=0$. Integrate to find original function.
 
+## Solids of revolution
+
+Approximate as sum of infinitesimally-thick cylinders
+
+### Rotation about $x$-axis
+
+\begin{align*}
+  V &= \int^{x=b}_{x-a} \pi y^2 \> dx \\
+    &= \pi \int^b_a (f(x))^2 \> dx
+\end{align*}
+
+### Rotation about $y$-axis
+
+\begin{align*}
+  V &= \int^{y=b}_{y=a} \pi x^2 \> dy \\
+    &= \pi \int^b_a (f(y))^2 \> dy
+\end{align*}
+
+### Regions not bound by $y=0$
+
+$$V = \pi \int^b_a f(x)^2 - g(x)^2 \> dx$$  
+where $f(x) > g(x)$
+
 ## Rates
 
 ### Related rates