auto-update(nvim): 2025-03-03 00:43:58

documents/by-course/math-6a/course-notes/main.typ

@@ -1106,5 +1106,63 @@ Any 3D shape can be built recursively of atomic objects.
 
 Schley what are you doing???
 
-== Signed area and volume
+= Double integrals and some fun
 
+We consider various types of integrals. Consult
+#link("https://web.evanchen.cc/textbooks/poster-ints.pdf")[this chart] to
+classify them.
+
+Namely, we consider integrals of 0, 1, 2, and 3 dimensions, on scalar valued
+function from $RR^0 -> RR$, $RR^1 -> RR$, $RR^2 -> RR$, and $RR^3 -> RR$.
+
+To evaluate double integrals, we need to place notable importance on the
+_region_ we integrate over.
+
+== Integrating over rectangles
+
+Integrating over a rectangle is super easy. It's like partial derivatives,
+where we hold every other variable constant and integrate over our desired
+value.
+
+#example[
+  Evaluate
+
+  $ integral^6_0 integral^1_0 x y^2 dif x dif y $
+
+  We just crunch the numbers.
+
+  The first integral:
+  $
+    integral_0^1 x y^2 dif x = lr(1/2 x^2 y^2 |)_(x=0)^(x=1) = 1 / 2 y^2
+  $
+  The second integral:
+  $
+    integral_0^6 1 / 2 y^2 dif y = lr(1/6 y^3 |)_(y=0)^(y=6) = 36
+  $
+]
+
+== $x y$ integration without a rectangle
+
+In general most 2D regions $cal(R)$ can still be done with $x y$ integration even when they aren't rectangles. In that case, we use the notation
+$
+  integral.double_cal(R) f(x,y) dif x dif y := "integral of" f "over" cal(R)
+$
+If the region is given by inequalities, for instance, a unit disk, we would write
+$
+  integral.double_(x^2 + y^2 <= 1) f(x,y) dif x dif y
+$
+
+== Swapping the order of integration
+
+If our function is nice, then this is easy.
+
+#theorem[Fubini's][
+  If the function that defines our surface is continuous, the double integral
+  over a rectangle can be evaluated in either order without a change to the
+  integral.
+]
+
+Otherwise, we want to swap the order of integration. We convert the limits of
+integration back into inequality/region format, getting a region $cal(R)$ like
+discussed in the previous section. Then evaluate that integral using standard
+methods.

documents/by-course/pstat-120a/course-notes/main.typ

@@ -2261,7 +2261,7 @@ They are "easy" to use for finding the distributions of:
     E[X^k] = mu_k, k = 1,2,...
   $
   Then the *moment generating function* of a random variable $X$ is defined by
-  $M_x(t) = E[e^(t x)]$, for the real variable $t$.
+  $M_x (t) = E[e^(t x)]$, for the real variable $t$.
 ]
 
 All of the moments must be defined for the MGF to exist. The MGF looks like