some typos

quarto/integrals/area_between_curves.qmd

@@ -494,7 +494,7 @@ box(f⁻¹(x₀-1Δ), x₀-2Δ, 1 - f⁻¹(x₀-1Δ), Δ, colᵣ)
 box(f⁻¹(x₀-2Δ), x₀-3Δ, 1 - f⁻¹(x₀-2Δ), Δ, colᵣ)
 ```
 
-The figure above suggests that the area under $f(x)$ over $[a,b]$ could be represented as the area between the curves $f^{-1}(y)$ and $y=b$ from $[f(a), f(b)]$.
+The figure above suggests that the area under $f(x)$ over $[a,b]$ could be represented as the area between the curves $f^{-1}(y)$ and $x=b$ from $[f(a), f(b)]$.
 
 
 ---
@@ -552,7 +552,7 @@ p = plot(xs, ys; line=(3, :black), ylims=(0,4), legend=false)
 scatter!(p, xs,  ys; marker=(7, :circle))
 ```
 
-Going further, we draw the four trapezoids using different colors depending on the sign of the `xs[i+1] - xs[[i]` terms:
+Going further, we draw the four trapezoids using different colors depending on the sign of the `xs[i+1] - xs[i]` terms:
 
 ```{julia}
 for i in 1:4
@@ -563,7 +563,7 @@ end
 p
 ```
 
-The yellow trapezoids appear to be colored green, as they completely overlap with parts of the blue trapezoids and blue and yellow make green. As the signs of the differences of the $x$ values is different, these areas add to $0$ in the sum, leaving just the area of the interior when the sum is computed.
+The yellow trapezoids appear to be colored grey, as they completely overlap with parts of the blue trapezoids and blue and yellow make grey with lights. As the signs of the differences of the $x$ values is different, these areas add to $0$ in the sum, leaving just the area of the interior when the sum is computed.
 
 For this particular figure, the enclosed area is
 
@@ -964,9 +964,10 @@ end
 What does this imply:
 
 ```{julia}
+#| hold: true
 #| echo: false
 choices = ["The two enclosed areas should be equal",
-           "The two enclosed areas are clearly different, as they do not overap"],
+           "The two enclosed areas are clearly different, as they do not overap"]
 radioq(choices, 1)
 ```
-           "
+