From e8f494d2565d3be64a607fe00c99fd1176e521d0 Mon Sep 17 00:00:00 2001 From: eggy Date: Thu, 12 Jan 2023 11:17:26 -0500 Subject: [PATCH] ece106: add integration tips --- docs/1b/ece106.md | 23 ++++++++++++++++++++++- 1 file changed, 22 insertions(+), 1 deletion(-) diff --git a/docs/1b/ece106.md b/docs/1b/ece106.md index 119d24d..c8c5a1f 100644 --- a/docs/1b/ece106.md +++ b/docs/1b/ece106.md @@ -65,10 +65,31 @@ In polar form, the difference in each "rectangle" side length is slightly differ | $\hat r$ | $dr$ | | $\hat\phi$ | $rd\phi$ | -Therefore, the change in surface area is equal to: +Therefore, the change in surface area can be approximated to be a rectangle and is equal to: $$dS=(dr)(rd\phi)$$ +!!! example + The area of a circle can be expressed as $A=\int^{2\pi}_0\int^R_0 r\ dr\ d\phi$. + + $$ + \begin{align*} + A&=\int^{2\pi}_0\frac{1}{2}R^2\ d\phi \\ + &=\pi R^2 + \end{align*} + $$ + +So long as the variables are independent of each other, their order does not matter. Otherwise, the dependent variable must be calculated first. + + +!!! tip + There is a shortcut for integrals of cosine and sine squared, **so long as $a=0$ and $b$ is a multiple of $\frac\pi 2$**: + + $$ + \int^b_a\cos^2\phi=\frac{b-a}{2} \\ + \int^b_a\sin^2\phi=\frac{b-a}{2} + $$ + ## Cartesian coordinates The axes in a Cartesian coordinate plane must be orthogonal so that increasing a value in one axis does not affect any other. The axes must also point in directions that follow the **right hand rule**.