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quarto/integrals/ftc.qmd
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@@ -156,10 +156,75 @@ In Part 1, the integral $F(x) = \int_a^x f(u) du$ is defined for any Riemann int
:::
This figure relating the area under some continuous $f(x)$ from $a$ to both $x$ and $x+h$ for some small $h$ helps to visualize the two fundamental theorems.
::: {#fig-FTC-derivative}
```{julia}
#| echo: false
let
gr()
f(x) = sin(x)
A(x) = cos(x)
a,b = 0, 6pi/13
h = pi/20
xs = range(a, b, 100)
p1 = plot(; empty_style...)
plot!([0,0] .- 0.05,[-0.1, 1]; line=(:gray,1), arrow=true, side=:head)
plot!([-0.1, b+h + pi/10], [0,0]; line=(:gray,1), arrow=true, side=:head)
xs = range(a, b, 100)
S = Shape(vcat(xs, reverse(xs)), vcat(f.(xs), zero.(xs)))
plot!(S; fill=(:gray90, 0.25), line=(nothing,))
plot!(f, a, b+h; line=(:black, 2))
xs = range(b, b+h, 100)
S = Shape(vcat(xs, reverse(xs)), vcat(f.(xs), zero.(xs)))
plot!(S; fill=(:gray70, 0.25), line=(nothing,))
plot!([b,b,b+h,b+h],[0,f(b),f(b),0]; line=(:black,1,:dash))
annotate!([
(a,0,text(L"a", :top, :left)),
(b, 0, text(L"x", :top)),
(b+h,0,text(L"x+h", :top)),
(2b/3, 1/2, text(L"A(x)")),
(b + h/2, 1/2, text(L"f(x)\cdot h \approx A(x+h)-A(x)", rotation=90))
])
current()
end
```
```{julia}
#| echo: false
plotly()
nothing
```
Area under curve between $a$ and $b$ labeled with $A(b)$ for $b=x$ and $b=x+h$.
:::
The last rectangle is exactly $f(x)h$ and approximately $A(x+h)-A(x)$, the difference being the small cap above the shaded rectangle. This gives the approximate derivative:
$$
A'(x) \approx \frac{A(x+h) - A(x)}{h} \approx \frac{f(x)\cdot h}{h} = f(x)
$$
That is, by taking limits, $A(x) = \int_a^x f(u) du$ is an antiderivative of $f(x)$. Moreover, from geometric considerations of area, if $a < c < b$, then
$$
A(b) - A(c) = \int_a^b f(x) dx - \int_a^c f(x) dx = \int_c^b f(x) dx
$$
That is $A(x)$ satisfies the two parts of the fundamental theorem.
## Using the fundamental theorem of calculus to evaluate definite integrals
The major use of the FTC is the computation of $\int_a^b f(x) dx$. Rather than resort to Riemann sums or geometric arguments, there is an alternative - *when possible*, find a function $F$ with $F'(x) = f(x)$ and compute $F(b) - F(a)$.
The most visible use of the FTC is the computation of definite integrals, $\int_a^b f(x) dx$. Rather than resort to Riemann sums or geometric arguments, there is an alternative - *when possible*, find a function $F$ with $F'(x) = f(x)$ and compute $F(b) - F(a)$.
Some examples:
@@ -213,21 +278,21 @@ The expression $F(b) - F(a)$ is often written in this more compact form:
$$
\int_a^b f(x) dx = F(b) - F(a) = F(x)\big|_{x=a}^b, \text{ or just expr}\big|_{x=a}^b.
\int_a^b f(x) dx = F(b) - F(a) = F(x)\Big|_{x=a}^b, \text{ or just expr}\Big|_{x=a}^b.
$$
The vertical bar is used for the *evaluation* step, in this case the $a$ and $b$ mirror that of the definite integral. This notation lends itself to working inline, as we illustrate with this next problem where we "know" a function "$F$", so just express it "inline":
$$
\int_0^{\pi/4} \sec^2(x) dx = \tan(x) \big|_{x=0}^{\pi/4} = 1 - 0 = 1.
\int_0^{\pi/4} \sec^2(x) dx = \tan(x) \Big|_{x=0}^{\pi/4} = 1 - 0 = 1.
$$
A consequence of this notation is:
$$
F(x) \big|_{x=a}^b = -F(x) \big|_{x=b}^a.
F(x) \Big|_{x=a}^b = -F(x) \Big|_{x=b}^a.
$$
This says nothing more than $F(b)-F(a) = -F(a) - (-F(b))$, though more compactly.
@@ -324,13 +389,13 @@ Answers may not be available as elementary functions, but there may be special f
integrate(x / sqrt(1-x^3), x)
```
The different cases explored by `integrate` are after the questions.
Different cases explored by `integrate` are mentioned after the questions.
## Rules of integration
There are some "rules" of integration that allow integrals to be re-expressed. These follow from the rules of derivatives.
There are some "rules" of integration that allow indefinite integrals to be re-expressed.
* The integral of a constant times a function:
@@ -353,7 +418,7 @@ $$
This follows immediately as if $F(x)$ and $G(x)$ are antiderivatives of $f(x)$ and $g(x)$, then $[F(x) + G(x)]' = f(x) + g(x)$, so the right hand side will have a derivative of $f(x) + g(x)$.
In fact, this more general form where $c$ and $d$ are constants covers both cases:
In fact, this more general form where $c$ and $d$ are constants covers both cases and referred to by the linearity of the integral:
$$
@@ -373,7 +438,7 @@ $$
\begin{align*}
\int (a_n x^n + \cdots + a_1 x + a_0) dx
&= \int a_nx^n dx + \cdots + \int a_1 x dx + \int a_0 dx \\
&= a_n \int x^n dx + \cdots + a_1 \int x dx + a_0 \int dx \\
&= a_n \int x^n dx + \cdots + a_1 \int x^1 dx + a_0 \int x^0 dx \\
&= a_n\frac{x^{n+1}}{n+1} + \cdots + a_1 \frac{x^2}{2} + a_0 \frac{x}{1}.
\end{align*}
$$
@@ -417,12 +482,14 @@ This seems like a lot of work, and indeed it is more than is needed. The followi
$$
\int_0^\pi 100 \sin(x) dx = 100(-\cos(x)) \big|_0^{\pi} = 100 \cos(x) \big|_{\pi}^0 = 100(1) - 100(-1) = 200.
\int_0^\pi 100 \sin(x) dx = 100(-\cos(x)) \Big|_0^{\pi} = 100 \cos(x) \Big|_{\pi}^0 = 100(1) - 100(-1) = 200.
$$
## The derivative of the integral
The relationship that $[\int_a^x f(u) du]' = f(x)$ is a bit harder to appreciate, as it doesn't help answer many ready made questions. Here we give some examples of its use.
@@ -433,12 +500,16 @@ $$
F(x) = \int_a^x f(u) du.
$$
The value of $a$ does not matter, as long as the integral is defined.
The value of $a$ does not matter, as long as the integral is defined. This $F$ satisfies the first fundamental theorem, as $F(a)=0$.
```{julia}
#| hold: true
#| echo: false
#| eval: false
##{{{ftc_graph}}}
gr()
function make_ftc_graph(n)
@@ -479,9 +550,9 @@ imgfile = tempname() * ".gif"
gif(anim, imgfile, fps = 1)
plotly()
ImageFile(imgfile, caption)
```
The picture for this, for non-negative $f$, is of accumulating area as $x$ increases. It can be used to give insight into some formulas:
#The picture for this, for non-negative $f$, is of accumulating area as $x$ increases. It can be used to give insight into some formulas:
```
For any function, we know that $F(b) - F(c) + F(c) - F(a) = F(b) - F(a)$. For this specific function, this translates into this property of the integral:
@@ -550,7 +621,7 @@ In probability theory, for a positive, continuous random variable, the probabili
For example, the exponential distribution with rate $1$ has $f(x) = e^{-x}$. Compute $F(x)$.
This is just $F(x) = \int_0^x e^{-u} du = -e^{-u}\big|_0^x = 1 - e^{-x}$.
This is just $F(x) = \int_0^x e^{-u} du = -e^{-u}\Big|_0^x = 1 - e^{-x}$.
The "uniform" distribution on $[a,b]$ has
@@ -1120,6 +1191,192 @@ answ = 2
radioq(choices, answ)
```
###### Question
The error function (`erf`) is defined in terms of an integral:
$$
\text{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x \exp(-t^2) dt, \quad{x \geq 0}
$$
The constant is chosen so that $\lim_{x \rightarrow \infty} \text{erf}(x) = 1$.
What is the derivative of $\text{erf}(x)$?
```{julia}
#| echo: false
choices = [L"\exp(-x^2)",
L"-2x \exp(-x^2)",
L"\frac{2}{\sqrt{\pi}} \exp(-x^2)"]
radioq(choices, 3; keep_order=true, explanation="Don't forget the scalar multiple")
```
Is the function $\text{erf(x)}$ *increasing* on $[0,\infty)$?
```{julia}
#| echo: false
choices = ["No",
"Yes, the derivative is positive on this interval",
"Yes, the derivative is negative on this interval",
"Yes, the derivative is increasing on this interval",
"Yes, the derivative is decreasing on this interval"]
radioq(choices, 2; keep_order=true)
```
Is the function $\text{erf(x)}$ *concave down* on $[0,\infty)$?
```{julia}
#| echo: false
choices = ["No",
"Yes, the derivative is positive on this interval",
"Yes, the derivative is negative on this interval",
"Yes, the derivative is increasing on this interval",
"Yes, the derivative is decreasing on this interval"]
radioq(choices, 5; keep_order=true)
```
For $x > 0$, consider the function
$$
F(x) = \frac{2}{\sqrt{\pi}} \int_{-x}^0 \exp(-t^2) dt
$$
Why is $F'(x) = \text{erf}'(x)$?
```{julia}
#| echo: false
choices = ["The integrand is an *even* function so the itegral from ``0`` to ``x`` is the same as the integral from ``-x`` to ``0``",
"This isn't true"]
radioq(choices, 1; keep_order=true)
```
Consider the function
$$
F(x) = \frac{2}{\sqrt{\pi}} \int_0^{\sqrt{x}} \exp(-t^2) dt, \quad x \geq 0
$$
What is the derivative of $F$?
```{julia}
#| echo: false
choices = [L"\exp(-x^2)",
L"\frac{2}{\sqrt{\pi}} \exp(-x^2)",
L"\frac{2}{\sqrt{\pi}} \exp(-x^2) \cdot (-2x)"]
radioq(choices, 3; keep_order=true, explanation="Don't forget to apply the chain rule, as ``F(x) = \\text{erf}(\\sqrt{x})``")
```
###### Question
Define two function through the integrals:
$$
\begin{align*}
S(x) &= \int_0^x \sin(t^2) dt\\
C(x) &= \int_0^x \cos(t^2) dt
\end{align*}
$$
These are called *Fresnel Integrals*.
A non-performant implementation might look like:
```{julia}
S(x) = first(quadgk(t -> sin(t^2), 0, x))
```
Define a similar function for $C(x)$ and them make a parametric plot for $0 \le t \le 5$.
Describe the shape.
```{julia}
#| echo: false
choices = ["It makes a lovely star shape",
"It makes a lovely spiral shape",
"It makes a lovely circle"]
radioq(choices, 2; keep_order=true)
```
What is the value of $S'(x)^2 + C'(x)^2$ when $x=\pi$?
```{julia}
#| echo: false
numericq(1)
```
###### Question
Define a function with parameter $\alpha \geq 1$ by:
$$
\gamma(x; \alpha) = \int_0^x \exp(-t) t^{\alpha-1} dt, \quad x > 0
$$
What is the ratio of $\gamma'(2; 3) / \gamma'(2; 4)$?
```{julia}
#| echo: false
df(x,alpha) = exp(-x)*x^(alpha -1)
numericq(df(2,3)/df(2,4))
```
###### Question
Define a function
$$
i(x) = \int_0^{x^2} \exp(-t) t^{1/2} dt
$$
What is the derivative if $i$?
```{julia}
#| echo: false
choices = [L"\exp(-x) x^{1/2}",
L"\exp(-x) x^{1/2} \cdot 2x",
L"\exp(-x^2) (x^2)^{1/2}",
L"\exp(-x^2) (x^2)^{1/2}\cdot 2x"]
radioq(choices, 4; keep_order=true)
```
###### Question
The function `sinint` from `SpecialFunctions` computes
$$
F(x) = \int_0^x \frac{\sin(t)}{t} dt = \int_0^x \phi(t) dt,
$$
Where we define $\phi$ above to be $1$ when $t=0$, so that it will be continuous over $[0,x]$.
A related integral might be:
$$
G(x) = \int_0^x \frac{\sin(\pi t)}{\pi t} dt = \int_0^x \phi(\pi t) dt
$$
As this is an integral involving a simple transformation of $\phi(x)$, we can see that $G(x) = (1/\pi) F(\pi x)$. What is the derivative of $G$?
```{julia}
#| echo: false
choices = [
L"\phi(x)",
L"\phi(\pi x)",
L"\pi \phi(\pi x)"
]
radioq(choices, 2; keep_order=true)
```
###### Question
@@ -1144,12 +1401,14 @@ radioq(choices, answ, keep_order=true)
Barrow presented a version of the fundamental theorem of calculus in a 1670 volume edited by Newton, Barrow's student (cf. [Wagner](http://www.maa.org/sites/default/files/0746834234133.di020795.02p0640b.pdf)). His version can be stated as follows (cf. [Jardine](http://www.maa.org/publications/ebooks/mathematical-time-capsules)):
Consider the following figure where $f$ is a strictly increasing function with $f(0) = 0$. and $x > 0$. The function $A(x) = \int_0^x f(u) du$ is also plotted. The point $Q$ is $f(x)$, and the point $P$ is $A(x)$. The point $T$ is chosen to so that the length between $T$ and $x$ times the length between $Q$ and $x$ equals the length from $P$ to $x$. ($\lvert Tx \rvert \cdot \lvert Qx \rvert = \lvert Px \rvert$.) Barrow showed that the line segment $PT$ is tangent to the graph of $A(x)$. This figure illustrates the labeling for some function:
Consider the following figure where $f$ is a strictly increasing function with $f(0) = 0$. and $x > 0$. The function $A(x) = \int_0^x f(u) du$ is also plotted with a dashed red line. The point $Q$ is $f(x)$, and the point $P$ is $A(x)$. The point $T$ is chosen to so that the length between $T$ and $x$ times the length between $Q$ and $x$ equals the length from $P$ to $x$. ($\lvert Tx \rvert \cdot \lvert Qx \rvert = \lvert Px \rvert$.) Barrow showed that the line segment $PT$ is tangent to the graph of $A(x)$. This figure illustrates the labeling for some function:
```{julia}
#| hold: true
#| echo: false
let
gr()
f(x) = x^(2/3)
x = 2
A(x) = quadgk(f, 0, x)[1]
@@ -1160,14 +1419,21 @@ P = A(x)
secpt = u -> 0 + P/(x-T) * (u-T)
xs = range(0, stop=x+1/4, length=50
)
p = plot(f, 0, x + 1/4, legend=false)
plot!(p, A, 0, x + 1/4, color=:red)
p = plot(f, 0, x + 1/4, legend=false, line=(:black,2))
plot!(p, A, 0, x + 1/4, line=(:red, 2,:dash))
scatter!(p, [T, x, x, x], [0, 0, Q, P], color=:orange)
annotate!(p, collect(zip([T, x, x+.1, x+.1], [0-.15, 0-.15, Q-.1, P], ["T", "x", "Q", "P"])))
annotate!(p, collect(zip([T, x, x+.1, x+.1], [0-.15, 0-.15, Q-.1, P], [L"T", L"x", L"Q", L"P"])))
plot!(p, [T-1/4, x+1/4], map(secpt, [T-1/4, x + 1/4]), color=:orange)
plot!(p, [T, x, x], [0, 0, P], color=:green)
p
p
end
```
```{julia}
#| echo: false
plotly()
nothing
```
The fact that $\lvert Tx \rvert \cdot \lvert Qx \rvert = \lvert Px \rvert$ says what in terms of $f(x)$, $A(x)$ and $A'(x)$?