updates

quarto/integrals/mean_value_theorem.qmd

@@ -66,14 +66,18 @@ $$
 \text{average} = \frac{1}{\pi-0} \int_0^\pi \sin(x) dx = \frac{1}{\pi} (-\cos(x)) \big|_0^\pi = \frac{2}{\pi}
 $$
 
-Visually, we have:
+Visually:
 
 
 ```{julia}
-plot(sin, 0, pi)
-plot!(x -> 2/pi)
+#| label: fig-integral-mean-value
+#| fig-cap: "Area under sine curve is equal to area of rectangle"
+plot(sin, 0, pi, legend=false, fill=(:forestgreen, 0.25, 0))
+plot!(x -> 2/pi,               fill=(:royalblue,   0.25, 0))
 ```
 
+In @fig-integral-mean-value the area under the sine curve ($2 = (-\cos(\pi)) - (-\cos(0))$) is equal to the area under the average (also $2 = 2/\pi \cdot \pi$).
+
 ##### Example
 
 
@@ -91,7 +95,7 @@ What is the average value of the function $e^{-x}$ between $0$ and $\log(2)$?
 
 $$
 \begin{align*}
-\text{average} = \frac{1}{\log(2) - 0} \int_0^{\log(2)} e^{-x} dx\\
+\text{average} &= \frac{1}{\log(2) - 0} \int_0^{\log(2)} e^{-x} dx\\
 &= \frac{1}{\log(2)} (-e^{-x}) \big|_0^{\log(2)}\\
 &= -\frac{1}{\log(2)} (\frac{1}{2} - 1)\\
 &= \frac{1}{2\log(2)}.
@@ -103,8 +107,8 @@ Visualizing, we have
 
 
 ```{julia}
-plot(x -> exp(-x), 0, log(2))
-plot!(x -> 1/(2*log(2)))
+plot(x -> exp(-x), 0, log(2), legend=false, fill=(:forestgreen, 0.25, 0))
+plot!(x -> 1/(2*log(2)),                    fill=(:royalblue, 0.25, 0))
 ```
 
 ## The mean value theorem for integrals