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

Bir roket motorunun aşırı sıcaklıklara dayanmasını sağlamak için, mühendisler çan şeklindeki nozuluna hassas bir ısı dayanıklılığı tabakası uygulamalıdır.Gerekli malzemenin tam miktarını tahmin etmek için toplam Devrim Yüzeyi Alanı hesaplanmasını gerektirir.Nozulun profilini iki boyutlu bir düzlemde kavisli bir fonksiyon olarak düşünün. Bu eğriyi x ekseni etrafında döndürmek, devrim 3D yüzeyini oluşturur.Kesin alanı hesaplamak için, eğri düz çizgilerle yaklaştırılmış küçük alt aralıklara bölünür. Her düz segment eksen etrafında dönerek dar bir konik bant oluşturur.Tek bir bandın yüzey alanı, çevresinin eğik yüksekliği ile çarpılmasıyla hesaplanır. Yarıçap, bant boyunca değiştiği için hesaplama ortalama bir yarıçapı kullanır.Aynı zamanda, eğik yükseklik, o segment boyunca eğrinin Yay Uzunluğu kullanılarak hesaplanır. Bu, nozulun eğimindeki dikliği açıklar.Son olarak, bu bantların alanlarının toplamı bir yaklaşım sağlar. Segmentler sonsuz daraldıkça, ortalama yarıçapı fonksiyonun değerine yakınır. Bu limit, toplamı belirli bir integrale dönüştürür ve kesin yüzey alanını verir.

dönel yüzeyin alanı

Area of a Surface of Revolution

D&#xF6;nel y&#xFC;zeyler, iki boyutlu bir e&#x11F;rinin bir eksen etraf&#x131;nda d&#xF6;nd&#xFC;r&#xFC;lmesiyle olu&#x15F;ur ve &#xFC;&#xE7; boyutlu bir &#x15F;ekli ortaya &#xE7;&#x131;kar&#x131;r. Bu kavram, roket nozulunun y&#xFC;zey alan&#x131;n&#x131;n belirlenmesi gibi m&#xFC;hendislik g&#xF6;revlerinde kullan&#x131;l&#x131;r; burada kesin hesaplamalar, homojen &#x131;s&#x131;ya dayan&#x131;kl&#x131; kaplamalar&#x131;n uygulanmas&#x131; i&#xE7;in kritik &#xF6;neme sahiptir. Bir e&#x11F;ri x ekseni etraf&#x131;nda d&#xF6;nd&#xFC;r&#xFC;ld&#xFC;&#x11F;&#xFC;nde, s&#xFC;rekli bir y&#xFC;zey olu&#x15F;turur ve bu y&#xFC;zeyin alan&#x131;, malzeme gereksinimlerini tahmin edebilmek i&#xE7;in do&#x11F;ru bi&#xE7;imde hesaplanmal&#x131;d&#x131;r.
Konik Bantlarla Yakla&#x15F;t&#x131;rma
Y&#xFC;zey alan&#x131;n&#x131; tahmin etmek i&#xE7;in e&#x11F;ri, &#xF6;nce k&#x131;sa aral&#x131;klar boyunca k&#xFC;&#xE7;&#xFC;k do&#x11F;rusal par&#xE7;alara ayr&#x131;l&#x131;r. Bu par&#xE7;alar eksen etraf&#x131;nda d&#xF6;nd&#xFC;r&#xFC;ld&#xFC;&#x11F;&#xFC;nde, dar konik bantlar olu&#x15F;ur. Her bir bant&#x131;n y&#xFC;zey alan&#x131;, ortalama &#xE7;evresi ile ilgili par&#xE7;an&#x131;n uzunlu&#x11F;unu ve dikli&#x11F;ini yans&#x131;tan e&#x11F;ik y&#xFC;ksekli&#x11F;in &#xE7;arp&#x131;m&#x131; al&#x131;narak yakla&#x15F;&#x131;k olarak hesaplan&#x131;r. Her bant i&#xE7;in ortalama yar&#x131;&#xE7;ap, aral&#x131;k i&#xE7;inde orijinal fonksiyonun de&#x11F;erlerinden belirlenir; e&#x11F;ik y&#xFC;kseklik ise e&#x11F;rinin yay uzunlu&#x11F;undan elde edilir.
Kesin De&#x11F;ere Yak&#x131;nsama
Par&#xE7;alar giderek darald&#x131;k&#xE7;a yakla&#x15F;t&#x131;rman&#x131;n do&#x11F;rulu&#x11F;u artar. Limitte, aral&#x131;k geni&#x15F;likleri s&#x131;f&#x131;ra yakla&#x15F;t&#x131;&#x11F;&#x131;nda toplam y&#xFC;zey alan&#x131; belirli integral ile ifade edilir. Bu integral, de&#x11F;i&#x15F;en yar&#x131;&#xE7;ap ile e&#x11F;rinin e&#x11F;imini birlikte ele alarak y&#xFC;zeyin ger&#xE7;ek geometrisini yans&#x131;t&#x131;r. Sonu&#xE7;, havac&#x131;l&#x131;k m&#xFC;hendisli&#x11F;inde termal koruma gibi kritik &#xF6;neme sahip uygulamalar i&#xE7;in gerekli olan kesin bir y&#xFC;zey alan&#x131; de&#x11F;eridir.

Dönel yüzeyin alanı

D&#xF6;nel y&#xFC;zeyler, iki boyutlu bir e&#x11F;rinin bir eksen etraf&#x131;nda d&#xF6;nd&#xFC;r&#xFC;lmesiyle olu&#x15F;ur ve &#xFC;&#xE7; boyutlu bir ...

Education

JoVE Core

Calculus

Kalkülüs

Unit

Birim

Application of Techniques of Integration

İntegral Alma Tekniklerinin Uygulanması

KEY TERMS AND CONCEPTS

TEMEL TERİMLER VE KAVRAMLAR

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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The arc length function measures the distance traveled along a smooth curve from a fixed starting point to a moving endpoint. It is useful when a straight-line estimate is not accurate enough. For curves that are continuous and differentiable, arc length gives a precise way to track length along the curve. To find this distance, the curve is broken into many small pieces. Each piece is treated like a short straight line. Its length depends on the horizontal change and the vertical change over...

area of a surface of revolution

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...

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Hydrostatic Force on Dam Walls with Integration

Hydrostatic force is the total force a fluid at rest exerts on a surface. For a horizontal surface at a fixed depth, the pressure is constant. It is found by multiplying fluid density, gravitational acceleration, and depth. A vertical dam wall is different because pressure increases with depth. The water above each point adds to the pressure below it. That means the force is not spread evenly across the wall. Integration helps measure this changing force. Pressure in a still fluid rises...

applications of integration to find centers of mass

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Center of Mass in Discrete and Continuous Systems

Center of mass can be found from rotational equilibrium and from integration. A balanced plank on a pivot shows the idea clearly. When the net torque about the pivot is zero, the system is in balance. Torque is the force on an object times its perpendicular distance from the pivot. At that point, the pivot lies at the center of mass. For a set of discrete point masses, the center of mass is the position where the masses balance by weight. Each mass contributes its mass times its position,...

theorem of pappus

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...

applications of integration to find consumer surplus

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Consumer surplus measures the economic gain buyers receive when they pay less than the highest price they are willing to pay. In microeconomics, this idea comes from the demand function, which links quantity to the price consumers will pay. Because willingness to pay usually falls as quantity rises, the demand curve slopes downward. When a product is sold at a fixed price, some buyers would have paid more for the first units they bought. The difference between that willingness to pay and the...

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Finding Probability with Area Under a PDF

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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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