WIP; better figures

quarto/integrals/area.qmd

@@ -190,6 +190,40 @@ Clearly for a given partition and choice of $c_i$, the above can be computed. Ea
 #| hold: true
 #| echo: false
 gr()
+
+function left_riemann(n)
+		empty_style = (xaxis=([], false),
+                    yaxis=([], false),
+                    framestyle=:origin,
+                    legend=false)
+	axis_style = (arrow=true, side=:head, line=(:gray, 1))
+	rectangle = (x, y, w, h) -> Shape(x .+ [0,w,w,0], y .+ [0,0,h,h])
+
+	f = x -> -(x+1/2)*(x-1)*(x-3) + 1
+	a, b= 1, 3
+
+	plot(; empty_style...)
+	plot!(f, a, b; line=(:black, 3))
+	plot!([a-.25, b+.25], [0,0]; axis_style...)
+	plot!([a-.1, a-.1], [-.25, .5 + f(a/2 +b/2)]; axis_style...)
+
+	Δ = (b-a)/n
+	for i ∈ 0:n-1
+		xᵢ = a + i*Δ
+		plot!(rectangle(xᵢ, 0, Δ, f(xᵢ)), opacity=0.5, color=:red)
+	end
+	area = round(sum(f(a + i*Δ)*Δ for i ∈ 0:n-1), digits=3)
+
+	annotate!([
+		(a, 0, text(L"a", :top)),
+		(b, 0, text(L"b", :top)),
+		(a, f(a/2+b/2), text("\$L_{$n} = $area\$", :left))
+	])
+
+	current()
+end
+
+#=
 rectangle(x, y, w, h) = Shape(x .+ [0,w,w,0], y .+ [0,0,h,h])
 function ₙ(j)
 	a = ("₋","","","₀","₁","₂","₃","₄","₅","₆","₇","₈","₉")
@@ -210,6 +244,7 @@ function left_riemann(n)
 	title!("L$(ₙ(n)) = $a")
 	p
 end
+=#
 
 anim = @animate for i ∈ (2,4,8,16,32,64)
     left_riemann(i)
@@ -631,19 +666,19 @@ Before continuing, we define a function to compute  Riemann sums for us with an
 
 ```{julia}
 #| eval: false
-riemann(f, a, b, n; method="right") = riemann(f, range(a,b,n+1); method=method)
 function riemann(f, xs; method="right")
-    Ms = (left = (f,a,b) -> f(a),
-          right= (f,a,b) -> f(b),
+    Ms = (left      = (f,a,b) -> f(a),
+          right     = (f,a,b) -> f(b),
           trapezoid = (f,a,b) -> (f(a) + f(b))/2,
-          simpsons  = (f,a,b) -> (c = a/2 + b/2; (1/6) * (f(a) + 4*f(c) + f(b))),
+          simpsons  = (f,a,b) -> (c = a/2 + b/2; (1/6) * (f(a) + 4*f(c) + f(b)))
           )
-    _riemann(Ms[Symbol(method)], f, xs)
-end
-function _riemann(M, f, xs)
+    M = Ms[Symbol(method)}
     xs′ = zip(xs[1:end-1], xs[2:end])
     sum(M(f, a, b) * (b-a) for (a,b) ∈ xs′)
 end
+
+riemann(f, a, b, n; method="right") =
+    riemann(f, range(a,b,n+1); method)
 ```
 
 (This function is defined in `CalculusWithJulia` and need not be copied over if that package is loaded.)
@@ -699,7 +734,7 @@ Consider a function $g(x)$ defined through its piecewise linear graph:
 ```{julia}
 #| echo: false
 g(x) = abs(x) > 2 ? 1.0 : abs(x) - 1.0
-plot(g, -3,3)
+plot(g, -3,3; legend=false)
 plot!(zero)
 ```
 
@@ -710,6 +745,24 @@ plot!(zero)
 
 We could add the signed area over $[0,1]$ to the above, but instead see a square of area $1$, a triangle with area $1/2$ and a triangle with signed area $-1$. The total is then $1/2$.
 
+This figure may make the above decomposition more clear:
+
+```{julia}
+#| echo: false
+let
+	g(x) = abs(x) > 2 ? 1.0 : abs(x) - 1.0
+	xs = [ -3, -2, -1, 1, 2, 3]
+	plot(; legend=false, aspect_ratio=:equal)
+	plot!(Shape([-3,-2,-2,-3], [0,0,1,1]); fill=(:gray,))
+	plot!(Shape([-2,-1,-1,-2], [0,0,0,1]); fill=(:gray10,))
+	plot!(Shape([-1,0,1], [0,-1,0]); fill=(:gray90,))
+	plot!(Shape([1,2,2,1], [0,1,0,0]); fill=(:gray10,))
+	plot!(Shape([2,3,3,2], [0,0,1,1]); fill=(:gray,))
+end
+```
+
+
+
 
   * Compute $\int_{-3}^{3} g(x) dx$: