From 47c0b7b99a16b987078cc12e09e2da89c3310e6d Mon Sep 17 00:00:00 2001 From: eggy Date: Thu, 18 Feb 2021 18:49:57 -0500 Subject: [PATCH] math: add volume of revolution --- docs/mcv4u7.md | 13 +++++++++++++ 1 file changed, 13 insertions(+) diff --git a/docs/mcv4u7.md b/docs/mcv4u7.md index ff78203..441dd9e 100644 --- a/docs/mcv4u7.md +++ b/docs/mcv4u7.md @@ -120,6 +120,19 @@ $$A=\int^b_a [f(x)-g(x)]dx, f(x)\geq g(x)\text{ in } [a,b]$$ If the limits of integration are not given, they are the outermost points of intersection of the two curves. +### Volumes of solids of revolution + +Shapes formed by rotating a line or curve about a fixed axis, such as cones, spheres, and cylinders are all known as **solids of revolution**. By splicing each shape into infinitely small disks, the cylinder volume formula can be used to find the volume of the solid. +$$ +\begin{align*} +V&=\lim_{x\to 0}\sum^b_{x=a}\pi y^2 dx \\ +&=\int^b_a y^2 dx +\end{align*} +$$ + +The area between two curves can also be rotated to form a solid, in which case its formula is: +$$V=\pi\int^b_a \big[g(x)^2-f(x)^2\big]dx, g(x)>f(x)$$ + ## Resources - [IB Math Analysis and Approaches Syllabus](/resources/g11/ib-math-syllabus.pdf)