From e960913f8fbaf6b668de83fd80178c4196d2d500 Mon Sep 17 00:00:00 2001 From: eggy Date: Sat, 19 Nov 2022 01:15:07 -0500 Subject: [PATCH] math117: add really shitty solids of revolution --- docs/ce1/math117.md | 24 ++++++++++++++++++++++++ 1 file changed, 24 insertions(+) diff --git a/docs/ce1/math117.md b/docs/ce1/math117.md index 51c88a3..91d3396 100644 --- a/docs/ce1/math117.md +++ b/docs/ce1/math117.md @@ -642,6 +642,30 @@ $$L(y)=\int^b_a\sqrt{1+\left(\frac{dx}{dy}\right)^2\ dy}$$ ### Solids of revolution +Please see [SL Math - Analysis and Approaches 2#Volumes of solids of revolution](/g11/mcv4u7/#volumes-of-solids-of-revolution) for more information. + +The **parallel axis theorem can be used** to shift the axis of the solid to $y=k$: + +$$V=\pi\int^b_a [f(x)^2 + 2kf(x)]\ dx$$ + +Around the vertical axis about the origin with a function that is bounded by $y$: + +$$V=\int^b_a2\pixf(x)\ dx$$ + +Around the vertical axis about the origin with functions bounded by $x$: + +$$V=\int^b_a2\pi(x-k)[f(x)-g(x)]\ dx$$ + +The **frustrum** is the sesction bounded by two parallel plates. + +The surface area of the solids are as follows: + +$$SA=\int^b_a2\pi f(x)\sqrt{1+f'(x)^2}\ dx$$ + +Around the vertical axis about the origin: + +$$SA=\int^b_a2\pi x\sqrt{1+f'(x)^2}\ dx$$ + ### Improper integrals An improper integral is a definite integral where only one bound is defined: