Name: calculating surface area from rotation
Uploaded: 2026-01-12T12:24:30.000Z
Duration: 1 min 29 s
Channel: JoVE

ロケットエンジンが極端な高温に耐えられるようにするために、エンジニアはベル型ノズルに正確な耐熱コーティング層を塗布しなければなりません。必要な材料の正確な量を見積もるには、回転面積の総表面積を計算する必要があります。ノズルのプロファイルを二次元平面上の曲面関数と考えます。この曲線をx軸を回転させることで3次元の回転面が生成されます。正確な面積を計算するために、曲線は直線で近似された小さな部分区間に分割されます。各直線の区間を軸の周りに回転させることで、狭い円錐形の帯を形成します。単一帯の表面積は、その周囲長に傾斜高さを掛けて計算されます。半径は帯域内で変動するため、計算では平均半径が用いられます。同時に、その線分に沿った曲線の弧長を用いて傾斜高さを計算します。これがノズルの傾斜の急勾配の理由です。最後に、これらの帯の面積を合計することで近似が得られます。セグメントが無限に狭くなると、平均半径は関数の値に収束します。この極限は和を定積分に変換し、正確な表面積を与えます。

回転面の面積

Area of a Surface of Revolution

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Education

JoVE Core

Calculus

微積分学

Unit

単位

Application of Techniques of Integration

積分法の応用

KEY TERMS AND CONCEPTS

主要用語と概念

calculating surface area from rotation

Surfaces of revolution are created when a two-dimensional curve is rotated around an axis. The rotation produces a three-dimensional shape. In engineering, this calculation can help determine the surface area of a rocket nozzle. That information matters when a heat-resistant coating must be applied evenly.

When a curve is revolved about the x-axis, it sweeps out a continuous surface. To find its area, the curve is first broken into small straight-line pieces over short intervals. Each rotated piece forms a narrow conical band, which is a thin ring-like section of a cone. The area of each band is estimated using its average circumference and its slant height, which is the tilted length of that segment.

The radius for each band comes from the function values over the interval. The slant height comes from the arc length of the curve. As the intervals become smaller, the estimate becomes more accurate. In the limit, the surface area is written as a definite integral.

This integral includes the changing radius and the slope of the curve. It captures the true shape of the surface instead of a rough estimate. The final result gives a precise surface area value. That precision is important in high-stakes work such as thermal protection for aerospace parts.

Calculating Surface Area from Rotation

arc length of a curve

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area of a surface of revolution

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Center of Mass in Discrete and Continuous Systems

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Pappus Theorem for Volume and Surface Area

Pappus Theorem connects centroid motion to the volume and surface area of solids of revolution. It is also called the Pappus–Guldinus Theorem. The method uses geometry to find these values when a plane region or a plane curve is rotated about an external axis. The theorem has two related parts. The first part gives the volume of a solid formed by rotating a plane region around an axis outside the region. In this case, the volume equals the area of the region multiplied by the distance traveled...

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key terms and concepts

Learn how rotating a curve around the x-axis creates a surface and how conical bands lead to a definite integral for exact surface area.

Calculating Surface Area from Rotation | 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.

calculus

Calculating Surface Area from Rotation

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