daviddoji/CalculusWithJuliaNotes.jl
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some typos

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Fang Liu
2025-06-06 16:56:56 +08:00
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quarto/integrals/area.qmd
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@@ -17,7 +17,7 @@ using Roots
---
The question of area has long fascinated human culture. As children, we learn early on the formulas for the areas of some geometric figures: a square is $b^2$, a rectangle $b\cdot h$ a triangle $1/2 \cdot b \cdot h$ and for a circle, $\pi r^2$. The area of a rectangle is often the intuitive basis for illustrating multiplication. The area of a triangle has been known for ages. Even complicated expressions, such as [Heron's](http://tinyurl.com/mqm9z) formula which relates the area of a triangle with measurements from its perimeter have been around for 2000 years. The formula for the area of a circle is also quite old. Wikipedia dates it as far back as the [Rhind](http://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrus) papyrus for 1700 BC, with the approximation of $256/81$ for $\pi$.
The question of area has long fascinated human culture. As children, we learn early on the formulas for the areas of some geometric figures: a square is $b^2$, a rectangle $b\cdot h$, a triangle $1/2 \cdot b \cdot h$ and for a circle, $\pi r^2$. The area of a rectangle is often the intuitive basis for illustrating multiplication. The area of a triangle has been known for ages. Even complicated expressions, such as [Heron's](http://tinyurl.com/mqm9z) formula which relates the area of a triangle with measurements from its perimeter have been around for 2000 years. The formula for the area of a circle is also quite old. Wikipedia dates it as far back as the [Rhind](http://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrus) papyrus for 1700 BC, with the approximation of $256/81$ for $\pi$.
The modern approach to area begins with a non-negative function $f(x)$ over an interval $[a,b]$. The goal is to compute the area under the graph. That is, the area between $f(x)$ and the $x$-axis between $a \leq x \leq b$.
@@ -480,9 +480,9 @@ This is just the area of a trapezoid with heights $a$ and $b$ and side length
$$
\begin{align*}
S &= x_1 \cdot (x_1 - x_0) + x_2 \cdot (x_2 - x_1) + \cdots x_n \cdot (x_n - x_{n-1}) \\
&= (a + 1\frac{b-a}{n}) \cdot \frac{b-a}{n} + (a + 2\frac{b-a}{n}) \cdot \frac{b-a}{n} + \cdots (a + n\frac{b-a}{n}) \cdot \frac{b-a}{n}\\
&= n \cdot a \cdot (\frac{b-a}{n}) + (1 + 2 + \cdots n) \cdot (\frac{b-a}{n})^2 \\
S &= x_1 \cdot (x_1 - x_0) + x_2 \cdot (x_2 - x_1) + \cdots + x_n \cdot (x_n - x_{n-1}) \\
&= (a + 1\frac{b-a}{n}) \cdot \frac{b-a}{n} + (a + 2\frac{b-a}{n}) \cdot \frac{b-a}{n} + \cdots + (a + n\frac{b-a}{n}) \cdot \frac{b-a}{n}\\
&= n \cdot a \cdot (\frac{b-a}{n}) + (1 + 2 + \cdots + n) \cdot (\frac{b-a}{n})^2 \\
&= n \cdot a \cdot (\frac{b-a}{n}) + \frac{n(n+1)}{2} \cdot (\frac{b-a}{n})^2 \\
& \rightarrow a \cdot(b-a) + \frac{(b-a)^2}{2} \\
&= \frac{b^2}{2} - \frac{a^2}{2}.
@@ -729,7 +729,7 @@ An immediate consequence would be $\int_{-\pi}^\pi \sin(x) = 0$, as would $\int_
##### Example
Numerically estimate the definite integral $\int_0^e x\log(x) dx$. (We redefine the function to be $0$ at $0$, so it is continuous.)
Numerically estimate the definite integral $\int_0^2 x\log(x) dx$. (We redefine the function to be $0$ at $0$, so it is continuous.)
We have to be a bit careful with the Riemann sum, as the left Riemann sum will have an issue at $0=x_0$ (`0*log(0)`) returns `NaN` which will poison any subsequent arithmetic operations, so the value returned will be `NaN` and not an approximate answer. We could define our function with a check:
@@ -860,7 +860,7 @@ $$
\frac{b-a}{6}(f(x_1) + 4f(x_2) + f(x_3)).
$$
This formula will actually be exact for any 3rd degree polynomial. In fact an entire family of similar approximations using $n$ points can be made exact for any polynomial of degree $n-1$ or lower. But with non-evenly spaced points, even better results can be found.
This formula will actually be exact for any 2nd degree polynomial. In fact an entire family of similar approximations using $n$ points can be made exact for any polynomial of degree $n-1$ or lower. But with non-evenly spaced points, even better results can be found.
The formulas for an approximation to the integral $\int_{-1}^1 f(x) dx$ discussed so far can be written as: