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7.2 Calculus of Parametric Curves

3 min read•june 24, 2024

offer a powerful way to describe complex paths in two dimensions. By using separate functions for x and y coordinates, we can model intricate shapes and motions that would be difficult to express with a single equation.

These curves open up new possibilities for calculus. We can find , areas, arc lengths, and surface areas of using specialized formulas tailored to . This approach lets us tackle a wider range of real-world problems.

Parametric Curves

Derivatives of parametric curves

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  • Define a curve using two functions [x(t)](https://www.fiveableKeyTerm:x(t))[x(t)](https://www.fiveableKeyTerm:x(t)) and [y(t)](https://www.fiveableKeyTerm:y(t))[y(t)](https://www.fiveableKeyTerm:y(t)) where tt is the
  • Find the derivative using the formula dydx=dy/dtdx/dt=y′(t)x′(t)\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{y'(t)}{x'(t)}
  • equation at a point (x(t0),y(t0))(x(t_0), y(t_0)) is y−y(t0)=y′(t0)x′(t0)(x−x(t0))y - y(t_0) = \frac{y'(t_0)}{x'(t_0)}(x - x(t_0))
  • Find points where the tangent line is horizontal by setting y′(t)=0y'(t) = 0 ()
  • Find points where the tangent line is vertical by setting x′(t)=0x'(t) = 0 ()

Area within parametric curves

  • Find the enclosed by a parametric curve using the formula A=∫t1t2y(t)x′(t)dtA = \int_{t_1}^{t_2} y(t)x'(t)dt
  • Works for curves traced out only once ()
  • If the curve is traced out multiple times, divide the area by the number of times the curve is traced ()

Arc length of parametric curves

  • Calculate the of a parametric curve from t=at = a to t=bt = b using the formula L=∫ab(x′(t))2+(y′(t))2dtL = \int_a^b \sqrt{(x'(t))^2 + (y'(t))^2}dt
  • Derived using the Pythagorean theorem and the concept of infinitesimal arc lengths ()
  • Simplify the integral by expressing the integrand in terms of a single variable, either xx or yy ()

Surface area from parametric revolution

  • Find the of a solid formed by revolving a parametric curve around the xx-axis using S=2π∫aby(t)(x′(t))2+(y′(t))2dtS = 2\pi \int_a^b y(t)\sqrt{(x'(t))^2 + (y'(t))^2}dt
  • Find the surface area of a solid formed by revolving a parametric curve around the yy-axis using S=2π∫abx(t)(x′(t))2+(y′(t))2dtS = 2\pi \int_a^b x(t)\sqrt{(x'(t))^2 + (y'(t))^2}dt
  • Derived using the concept of surface area as the sum of infinitesimal surface patches ()
  • Set up the integral using the correct limits of integration based on the given parametric equations ()

Vector-valued functions and curvature

  • Represent parametric curves as vector-valued functions in the form r(t)=⟨x(t),y(t)⟩\mathbf{r}(t) = \langle x(t), y(t) \rangle
  • Calculate the of a parametric curve using the formula κ=∣x′y′′−y′x′′∣(x′2+y′2)3/2\kappa = \frac{|x'y'' - y'x''|}{(x'^2 + y'^2)^{3/2}}
  • Find the to a parametric curve, which is perpendicular to the tangent vector and points towards the center of curvature

Applications and Examples

Solve problems involving real-world applications of parametric curves

  • Model real-world phenomena using parametric equations
    1. Motion of objects along a curved path ()
    2. Planetary orbits ()
    3. Cycloids (the path traced by a point on a rolling circle)
  • Solve application problems by
    1. Identifying the relevant parametric equations
    2. Determining the appropriate formula based on the question (, area, or surface area)
    3. Setting up and evaluating the integral, paying attention to the limits of integration
  • Calculate the distance traveled by a particle moving along a parametric curve ()
  • Find the area enclosed by the path of a planetary orbit ()
  • Determine the surface area of a solid formed by revolving a parametric curve representing a real-world object ()
  • Convert between parametric equations and for certain types of curves (e.g., spirals)

Key Terms to Review (37)

Arc length: Arc length is the distance measured along the curve between two points. It is calculated by integrating the square root of the sum of the squares of derivatives of the function defining the curve.
Arc Length: Arc length is the distance measured along a curved line or path, typically in the context of calculus and geometry. It represents the length of a segment of a curve, and is an important concept in understanding the behavior and properties of various mathematical functions and their graphical representations.
Area: Area is a measure of the size or extent of a two-dimensional surface or region. It quantifies the amount of space occupied by a shape or object within a plane.
Astroid: An astroid is a special type of parametric curve that resembles a four-lobed rose shape. It is defined by a set of parametric equations and has interesting geometric properties that are useful in the study of parametric curves.
Catenary: A catenary is the curve formed by a perfectly flexible chain suspended by its ends and acted on by gravity. Mathematically, it is described by the hyperbolic cosine function.
Catenary: A catenary is the curve formed by a uniform, flexible chain or cable suspended from two fixed points. It is the shape that a hanging chain or cable naturally assumes under the influence of gravity.
Chain Rule: The chain rule is a fundamental concept in calculus that allows for the differentiation of composite functions. It provides a systematic way to find the derivative of a function that is composed of other functions.
Curtate cycloid: A curtate cycloid is a type of curve generated by a point on the interior of a circle as it rolls along a straight line. It differs from a regular cycloid in that the tracing point is not on the circumference but inside the circle.
Curvature: Curvature is a measure of how much a curve deviates from a straight line at a given point. It describes the rate of change in the direction of a curve, providing information about the shape and bending of the curve.
Cycloid: A cycloid is a geometric curve that is traced by a point on the circumference of a circle as it rolls along a straight line. It is a fundamental concept in the study of parametric equations and the calculus of parametric curves.
Derivatives: Derivatives are functions that describe the rate of change of another function with respect to one or more of its independent variables. They are a fundamental concept in calculus that allow for the analysis of how a function's output changes as its input changes.
Dy/dx: The derivative, or dy/dx, represents the rate of change of a function y with respect to the independent variable x. It is a fundamental concept in calculus that describes the slope or instantaneous rate of change of a function at a particular point.
Elliptical Orbits: Elliptical orbits refer to the paths traced by objects in space, such as planets, moons, and satellites, around a central body under the influence of gravity. These orbits take the shape of an ellipse, a closed curve with two focal points, rather than a perfect circle.
Helix: A helix is a three-dimensional geometric shape that follows a spiral path, resembling the structure of a coiled spring or a twisted ribbon. It is a fundamental concept in various fields, including mathematics, physics, and biology, and is particularly relevant in the context of calculus of parametric curves.
Horizontal Tangent: A horizontal tangent is a point on a curve where the tangent line is parallel to the x-axis, indicating that the slope of the curve at that point is zero. This means the rate of change of the function is zero at that point, and the curve is neither increasing nor decreasing.
Kepler's Second Law: Kepler's second law, also known as the law of equal areas, states that a planet sweeps out equal areas in equal intervals of time as it orbits the Sun. This means that the line connecting a planet to the Sun sweeps out equal areas in equal amounts of time, regardless of the planet's position in its orbit.
Limaçon: A limaçon is a type of plane curve that resembles the shape of a snail's shell. It is a particular type of parametric curve that is often studied in the context of calculus of parametric curves.
Normal Vector: The normal vector of a parametric curve is a vector that is perpendicular to the tangent vector of the curve at a given point. It represents the direction that is normal, or orthogonal, to the direction of the curve at that point.
Parameter: A parameter is a variable that is used to describe a set of equations, often defining a curve or surface. It allows for the expression of coordinates as functions of one or more independent variables.
Parameter: A parameter is a variable that serves as an input to a function, equation, or model, and helps to define the specific characteristics or behavior of that mathematical construct. Parameters are used to provide flexibility and control over the output or results generated by these mathematical representations.
Parametric curves: Parametric curves are representations of curves in which the coordinates of the points on the curve are expressed as functions of a variable, typically denoted as 't'. This approach allows for the description of complex shapes and motion in a more flexible manner than traditional Cartesian coordinates, facilitating calculations like arc length and surface area. The equations can illustrate various geometric properties and behaviors over time, providing a powerful tool in calculus and physics.
Parametric Equations: Parametric equations are a set of equations that express the coordinates of points on a curve as functions of a variable, typically called the parameter. This approach allows for the representation of complex curves and shapes that might not be easily described by a single equation in Cartesian coordinates, thus making them useful in various mathematical applications, including determining arc lengths and surface areas, solving differential equations, and exploring polar coordinates.
Polar Coordinates: Polar coordinates are a two-dimensional coordinate system that uses a distance from a fixed point (the origin) and an angle to specify the location of a point. This system contrasts with the more common Cartesian coordinate system, which uses two perpendicular axes to define a point's position.
Projectile Motion: Projectile motion is the motion of an object that is launched into the air and moves under the influence of gravity and other forces, such as air resistance. It is a type of motion that follows a curved trajectory, with the object's path being determined by its initial velocity, launch angle, and the acceleration due to gravity.
Revolution: A revolution is the complete rotation or circular motion of an object around a fixed axis or point. It is a fundamental concept in various fields, including mathematics, physics, and engineering, where it is used to describe the motion of objects and the calculation of related quantities such as volume and arc length.
Roller Coaster: A roller coaster is an amusement park ride that consists of a track with a series of steep inclines and descents, often with loops and other thrilling maneuvers. It is a popular form of entertainment that provides an exhilarating and adrenaline-filled experience for riders through its dynamic motion and changing directions.
Simple Closed Curve: A simple closed curve is a continuous loop in the plane that does not intersect itself and returns to its starting point. It represents a closed path that encloses a finite region of the plane without any self-crossings.
Solid of revolution: A solid of revolution is a three-dimensional object obtained by rotating a two-dimensional region around an axis. The volume of such solids can be calculated using integration techniques.
Sphere: A sphere is a three-dimensional geometric shape that is perfectly round, with all points on the surface equidistant from the center. Spheres are fundamental in the study of calculus, particularly in the context of arc length, surface area, and parametric curves.
Surface Area: Surface area is the total area that the surface of a three-dimensional object occupies. It is crucial for understanding how objects interact with their environment, such as in calculating material requirements or heat transfer. The calculation of surface area is especially relevant when analyzing shapes formed by rotation, measuring lengths of curves, and examining parametrically defined shapes.
Tangent Line: A tangent line is a straight line that touches a curve at a single point, without crossing or intersecting the curve. It represents the instantaneous rate of change of the curve at that specific point, providing valuable information about the behavior and properties of the curve.
Torus: A torus is a three-dimensional geometric shape that resembles a doughnut or an inner tube. It is generated by revolving a circle around an axis that does not intersect the circle, creating a surface that has a hole in the middle.
Vase: A vase is a container, typically made of ceramic, glass, or metal, used to hold and display flowers or other decorative items. In the context of calculus of parametric curves, the term 'vase' refers to a specific type of parametric curve that resembles the shape of a vase.
Vector-Valued Function: A vector-valued function is a mathematical function that assigns a vector, rather than a scalar, to each input value. It maps elements from the domain to vectors in the codomain, representing a collection of related scalar functions.
Vertical Tangent: A vertical tangent is a point on a curve where the slope of the tangent line is vertical, meaning it is perpendicular to the x-axis. This occurs when the derivative of the function is undefined at that point, indicating a critical point where the function changes direction.
X(t): In the context of calculus of parametric curves, x(t) represents the x-coordinate of a point on a parametric curve as a function of the parameter t. It describes the horizontal position of the point as the curve is traced over time.
Y(t): y(t) is a function that represents the vertical or y-coordinate of a point on a parametric curve as a function of the parameter t. It is one of the key components in the study of calculus of parametric curves, as it allows for the analysis and understanding of the behavior of a curve in the y-direction.
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Glossary
Glossary
7.3 Polar Coordinates