Show description
Calculus II: A Guide to Parametric & Polar Coordinates
Calculus II: A Guide to Parametric & Polar Coordinates
Calculus II
A Guide to Parametric & Polar Coordinates
1. Parametric Curves
Instead of defining a path with $y$ in terms of $x$, we use a third variable, $t$ (think of it as time). A point's location is given by its coordinates $(x(t), y(t))$. This lets us trace complex paths, like the flight of a projectile or a path that crosses itself.
Finding the Cartesian Equation
To see the underlying path, we try to eliminate the parameter $t$.
Solve one of the equations for $t$.
Substitute that expression for $t$ into the other equation.
For trig functions, use identities like $\sin^2(t) + \cos^2(t) = 1$.
2. Calculus with Parametric Curves
Even with this new way of defining curves, we can still find slopes and lengths. We just need to adapt our formulas.
Slope of the Tangent Line
The slope $\frac{dy}{dx}$ is found by the ratio of how $y$ changes with respect to $t$ and how $x$ changes with respect to $t$.
Slope: $$\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$$
Arc Length
To find the length of a path from $t=a$ to $t=b$, we integrate the "speed" along the curve. This comes from a version of the Pythagorean theorem.
Arc Length: $$L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt$$
3. Polar Coordinates
Instead of a rectangular grid $(x, y)$, we navigate with a distance from the origin, $r$, and an angle, $\theta$. A point is given by $(r, \theta)$. This is perfect for describing things with circular or rotational symmetry.
Conversion Formulas
These are the formulas you need to translate between Cartesian and Polar coordinates.
From Polar to Cartesian:
$x = r \cos(\theta)$
$y = r \sin(\theta)$
From Cartesian to Polar:
$r^2 = x^2 + y^2 \implies r = \sqrt{x^2 + y^2}$
$\tan(\theta) = \frac{y}{x}$
Warning: When finding $\theta$ from $\tan(\theta) = y/x$, always check which quadrant your $(x,y)$ point is in to make sure you have the correct angle!
4. Areas and Lengths in Polar Coordinates
We can also measure area and length using polar coordinates.…
Calculus II: A Guide to Parametric & Polar Coordinates
<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<title>Calculus II: A Guide to Parametric & Polar Coordinates</title>
<script src="https://cdn.tailwindcss.com"></script>
<script>
MathJax = {
tex: {
inlineMath: [['$', '$'], ['\\(', '\\)']],
displayMath: [['$$', '$$'], ['\\[', '\\]']]
},
svg: {
fontCache: 'global'
}
};
</script>
<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
<link href="https://fonts.googleapis.com/css2?family=Playfair+Display:wght@700&family=Roboto:wght@400;500;700&display=swap" rel="stylesheet">
<style>
/* Maroon Theme */
body {
font-family: 'Roboto', sans-serif;
background-color: #3D0000; /* Dark Maroon */
color: #F5E6E0; /* Light parchment/cream text */
}
h1, h2, h3 {
font-family: 'Playfair Display', serif;
font-weight: 700;
}
.card {
background-color: #5C0000; /* Richer Maroon */
border-radius: 0.25rem;
box-shadow: 0 4px 15px rgba(0, 0, 0, 0.4);
margin-bottom: 2rem;
padding: 1.5rem;
border-left: 4px solid #FFD700; /* Gold accent */
}
.formula {
background-color: #3d1c1c; /* Darker, browner maroon */
border: 1px solid #800000; /* Maroon border */
padding: 1rem;
border-radius: 0.25rem;
margin: 1rem 0;
font-size: 1.1rem;
overflow-x: auto;
color: #f5e6e0;
}
.key-point {
font-weight: 700;
color: #FFD700; /* Gold */
}
.pitfall {
background-color: #6b4a4a; /* Muted red warning background */
border: 1px solid #FFBF00; /* Amber/Gold warning border */
padding: 1rem;
border-radius: 0.25rem;
color: #f5e6e0;
}
.pitfall strong {
color: #FFBF00; /* Amber/Gold */
}
mjx-container {
color: #f5e6e0 !important;
}
</style>
</head>
<body class="p-4 md:p-8">
<div class="max-w-4xl mx-auto">
<header class="text-center mb-12">
<h1 class="text-4xl md:text-5xl text-white">Calculus II</h1>
<p class="text-lg text-amber-200 mt-2">A Guide to Parametric & Polar Coordinates</p>
</header>
<!-- Parametric Curves -->
<div class="card">
<h2 class="text-3xl mb-4 text-amber-100">1. Parametric Curves</h2>
<p class="mb-4">Instead of defining a path with $y$ in terms of $x$, we use a third variable, $t$ (think of it as time). A point's location is given by its coordinates $(x(t), y(t))$. This lets us trace complex paths, like the flight of a projectile or a path that crosses itself.</p>
<h3 class="text-2xl mt-6 mb-2">Finding the Cartesian Equation</h3>
<p>To see the underlying path, we try to <span class="key-point">eliminate the parameter $t$</span>.
<ol class="list-decimal list-inside ml-4 mt-2">
<li>Solve one of the equations for $t$.</li>
<li>Substitute that expression for $t$ into the other equation.</li>
<li>For trig functions, use identities like $\sin^2(t) + \cos^2(t) = 1$.</li>
</ol>
</p>
</div>
<!-- Calculus with Parametric Curves -->
<div class="card">
<h2 class="text-3xl mb-4 text-amber-100">2. Calculus with Parametric Curves</h2>
<p class="mb-4">Even with this new way of defining curves, we can still find slopes and lengths. We just need to adapt our formulas.</p>
<h3 class="text-2xl mt-6 mb-2">Slope of the Tangent Line</h3>
<p>The slope $\frac{dy}{dx}$ is found by the ratio of how $y$ changes with respect to $t$ and how $x$ changes with respect to $t$.</p>
<div class="formula">
Slope: $$\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$$
</div>
<h3 class="text-2xl mt-6 mb-2">Arc Length</h3>
<p>To find the length of a path from $t=a$ to $t=b$, we integrate the "speed" along the curve. This comes from a version of the Pythagorean theorem.</p>
<div class="formula">
Arc Length: $$L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt$$
</div>
</div>
<!-- Polar Coordinates -->
<div class="card">
<h2 class="text-3xl mb-4 text-amber-100">3. Polar Coordinates</h2>
<p class="mb-4">Instead of a rectangular grid $(x, y)$, we navigate with a distance from the origin, $r$, and an angle, $\theta$. A point is given by $(r, \theta)$. This is perfect for describing things with circular or rotational symmetry.</p>
<h3 class="text-2xl mt-6 mb-2">Conversion Formulas</h3>
<p>These are the formulas you need to translate between Cartesian and Polar coordinates.</p>
<div class="formula">
<strong>From Polar to Cartesian:</strong>
<p class="mt-2">$x = r \cos(\theta)$</p>
<p>$y = r \sin(\theta)$</p>
</div>
<div class="formula">
<strong>From Cartesian to Polar:</strong>
<p class="mt-2">$r^2 = x^2 + y^2 \implies r = \sqrt{x^2 + y^2}$</p>
<p>$\tan(\theta) = \frac{y}{x}$</p>
</div>
<div class="pitfall mt-4">
<strong>Warning:</strong> When finding $\theta$ from $\tan(\theta) = y/x$, always check which quadrant your $(x,y)$ point is in to make sure you have the correct angle!
</div>
</div>
<!-- Areas and Lengths in Polar Coordinates -->
<div class="card">
<h2 class="text-3xl mb-4 text-amber-100">4. Areas and Lengths in Polar Coordinates</h2>
<p class="mb-4">We can also measure area and length using polar coordinates. The key is to think in terms of wedge-shaped sectors instead of rectangles.</p>
<h3 class="text-2xl mt-6 mb-2">Area of a Polar Region</h3>
<p>To find the area swept out by a polar curve $r = f(\theta)$ from angle $\alpha$ to $\beta$, we use the following integral.</p>
<div class="formula">
Area: $$A = \frac{1}{2} \int_\alpha^\beta [r(\theta)]^2 \, d\theta$$
</div>
<p class="mt-2">This is one of the most important formulas to remember for polar coordinates.</p>
<h3 class="text-2xl mt-6 mb-2">Arc Length in Polar</h3>
<p>This formula looks a bit like the parametric version, but adapted for polar variables $r$ and $\theta$.</p>
<div class="formula">
Arc Length: $$L = \int_\alpha^\beta \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} \, d\theta$$
</div>
</div>
</div>
</body>
</html>