Name: parametric curves and rotated surfaces
Uploaded: 2026-06-08T09:38:15.662Z
Duration: 1 min 30 s
Channel: JoVE

参数曲线定义了平面上的一条路径，其中 x 和 y 依赖于一个单一参数（通常是时间），该参数的取值范围从 alpha 到 beta。当曲线绕轴旋转时，会扫出一个三维曲面。这被称为旋转曲面。其表面积取决于曲线的形状及其到旋转轴的距离。考虑一个半径为 r 的圆，其中心位于距离 y 轴 R 处。该圆的 x 和 y 坐标由参数 t（从 0 到 2π）给出。将此圆绕同一平面内不与之相交的轴旋转，可形成一个环面。要以参数形式计算环面的表面积，需考虑其表面上的一个无穷小条带。当圆上的一个点旋转时，会描绘出一条圆形路径。该路径的周长即为条带的长度。条带的宽度则来源于参数曲线的弧长微分。将两者相乘得到微分面积，对该参数 t 进行积分即可得到参数形式下的总表面积。该方法有助于设计O型圈等部件，通过精确的表面积来确定其尺寸和接触面积，从而防止机器发生泄漏。

参数曲线的微积分：表面积

参数曲线是对平面上一条路径的描述，其中 x 和 y 坐标均为单一参数的函数，该参数通常记为 t。当这样的曲线绕同一平面内的一条外部轴旋转时，会形成一个三维的旋转曲面。该旋转体的表面积主要取决于两个方面：原始曲线的几何形状以及曲线到所选旋转轴的距离。环面是一种经典的旋转曲面，当一个圆绕其所在平面内且不与该圆相交的轴旋转时形成。考虑一个半径为 r 的圆，其圆心位于距离固定旋转轴 R 处。当该圆旋转时，其上的每一点都在空间中描绘出一个圆形路径。所形成的立体具有环状几何结构，并包含一个中心空腔；半径 r 控制环面管壁的厚度，而 R 决定环形“孔”的大小。从数学上讲，如果该圆位于 xy 平面内，且圆心在 (R, 0)，则可将其参数化为：x(t) = R + rcos t 和 y(t) = rsin t，其中 t 从 0 变化到 2𝜋。若将此曲线绕 y 轴旋转，即可生成环面。为了推导环面的表面积，需考虑原始曲线上的一个无穷小条带。该条带的宽度由弧长的微分给出，\begin{equation*}dS=\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\,dt\end{equation*}旋转时会扫出一个圆形“环”。该环到旋转轴的半径为 x(t)，因此条带扫过的圆形路径长度为 2𝜋x(t)。微分曲面面积元 dS 即为该路径长度与条带宽度的乘积。对参数 t 从 0 到 2𝜋 进行积分，可得总表面积：&#xA0;\begin{equation*}S= \int_{0}^{2\pi} 2\pi x(t)\,\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\,dt\end{equation*}对于环面，其表面积公式为：S = 4𝜋2Rr，这表明管半径 r 以及到轴线的距离 R 均会影响最终的表面积。

https://cloudfront.jove.com/files/ftp_upload/40605/40605_article_thumb_40605.jpg

calculus-with-parametric-curves-surface-areas

Education

教育

JoVE Core

Calculus

微积分

Unit

单元

Parametric Equations and Polar Coordinates

参数方程与极坐标

parametric-equations-and-polar-coordinates

KEY TERMS AND CONCEPTS

关键术语与概念

Calculus with Parametric Curves: Surface Areas

参数曲线微积分：表面积

parametric curves and rotated surfaces

Parametric curves can be used to find the surface area of a shape made by rotation. A parametric curve describes a path in the plane with both x and y as functions of one parameter, usually t. When that curve is revolved around an external axis in the same plane, it creates a surface of revolution in three dimensions.

The surface area depends on the shape of the curve and on how far the curve sits from the axis of rotation. A torus is a classic example of this kind of surface. It forms when a circle is rotated around an axis in its plane that does not cross the circle.

For a torus, the circle has radius r and its center is a distance R from the axis of rotation. As the circle turns, each point traces a circular path in space. The result is a ring-like solid with a hole in the middle. The radius r controls the thickness of the tube, and R controls the size of the hole.

If the circle lies in the xy-plane and is centered at (R, 0), it can be written as x(t) = R + r cos t and y(t) = r sin t, with t from 0 to 2π. Rotating this curve about the y-axis generates the torus. To find its surface area, one starts with a tiny strip on the curve and uses the arc length differential, dS = sqrt((dx/dt)^2 + (dy/dt)^2) dt.

When the strip rotates, it sweeps out a ring. The ring radius is x(t), so the path length is 2πx(t). The surface area element is the path length times the strip width. Integrating from 0 to 2π gives S = ∫₀²π 2πx(t)sqrt((dx/dt)^2 + (dy/dt)^2) dt.

For the torus, this integral becomes S = 4π²Rr. This result shows how both the tube radius r and the distance R from the axis affect the final surface area.

Parametric Curves and Rotated Surfaces

curves defined by parametric equations

Modeling Baseball Motion with Parametric Curves

Baseball motion can be modeled with parametric equations when air resistance is neglected. The path looks parabolic in a two-dimensional coordinate system, but the full motion is described as position changing with time. In this model, the horizontal position is written as f(t) and the vertical position as g(t). These functions give the x- and y-coordinates of the baseball at any time t. The horizontal part reflects constant horizontal velocity. The vertical part includes the initial vertical...

calculus with parametric curves: tangents and areas

calculus-with-parametric-curves-tangents-and-areas

Parametric Curves: Slope and Tangent Lines

Parametric curves use two functions, x(t) and y(t), to show the position of a point in the plane. The parameter t often stands for time. This setup helps describe both the path of the point and the direction it moves. The slope of a tangent line tells how steep the curve is at one point. For a parametric curve, the slope is found with dy/dx. Because x and y both depend on t, the chain rule gives the formula dy/dx = (dy/dt) / (dx/dt), as long as dx/dt is not zero. The derivatives also reveal...

calculus with parametric curves: surface areas

Parametric curves can be used to find the surface area of a shape made by rotation. A parametric curve describes a path in the plane with both x and y as functions of one parameter, usually t. When that curve is revolved around an external axis in the same plane, it creates a surface of revolution in three dimensions. The surface area depends on the shape of the curve and on how far the curve sits from the axis of rotation. A torus is a classic example of this kind of surface. It forms when a...

polar coordinate system

Mapping Points with Polar Coordinates

Polar coordinates map points in a plane by using distance and direction from a fixed center. This system is useful when a shape has circular or spiral symmetry, because those forms are easier to describe than they are on a rectangular grid. It can also help with regions that are not naturally rectangular. A ship’s plan position indicator is a familiar example. The ship sits at the center of the display, and detected targets appear as dots around it. In polar form, that center point is called...

polar curves

Spirograph Patterns in Polar Coordinates

Spirograph patterns show how polar coordinates can describe curves with angles and distances. Polar coordinates use a radius r, which is the distance from the origin, and an angle θ, measured counterclockwise from the polar axis. This system gives a useful alternative to Cartesian coordinates, which use x and y values. A polar curve is written as an equation in the form r = f(θ). As θ changes, the radius changes too, so each point moves farther from or closer to the origin. This makes polar...

integration applied to polar coordinates to find areas

integration-applied-to-polar-coordinates-to-find-areas

Polar Area of an Irregular Sprinkler Spray

Polar coordinates can be used to measure the area covered by an irregular sprinkler spray. In this setup, the spray distance changes with direction, so the radius is treated as a function of the angle of rotation. That changing radius creates a polar curve, which maps the sprinkler’s uneven reach across a full turn. To find the watered region, the area is split into many thin sectors. A sector is a narrow slice of a circle. Each slice has an area that depends on both its radius and the angle...

integration applied to polar coordinates to find arc lengths

integration-applied-to-polar-coordinates-to-find-arc-lengths

Polar Curve Distance with Calculus

Polar coordinates can be used to measure the distance along a curve that turns around a fixed point. In this system, a point is described by the radial distance r from the pole and the angle θ from a reference direction. That makes it useful for motion such as an expanding spiral, where distance and direction change together. A search drone gives a clear example of a polar path. If the hiker’s last known position is the pole, the drone’s location at any moment can be written as the polar...

polar coordinates: problem solving

Directional Beam Area in Polar Plots

Directional radiation patterns use polar plots to show how signal strength changes with direction. In a polar plot, the radial distance from the origin represents signal intensity at a given angle. This makes the graph useful for antenna analysis and for studying focused beams. A simple model for this kind of pattern is the four-lobed rose curve. It is described by r = cos(2θ) and has four symmetric lobes. Each lobe acts like a focused transmission beam in a simplified mathematical form. To...

key terms and concepts

Learn how parametric curves generate surfaces of revolution and how a rotated circle creates a torus. See the surface area setup and final formula.

Parametric Curves and Rotated Surfaces | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

Parametric Curves and Rotated Surfaces

Description

Transcript