ece106: add integration tips

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**.