Name: disk method for rotated region volume
Uploaded: 2026-01-12T12:24:09.000Z
Duration: 1 min 29 s
Channel: JoVE

Düzensiz şekilli nesnelerin hacimleri, devrim katı kavramı kullanılarak doğru şekilde belirlenebilir. Örneğin, iki boyutlu bir düzlemde eğri bir fonksiyon ele alalım. Bu eğriyi x ekseni etrafında döndürmek, devirli katı olarak bilinen simetrik üç boyutlu bir şekil oluşturur. Bölgenin etrafında dönen çizgi devrim eksenidir.Bölge doğrudan eksene karşı olduğunda, katı dairesel kesitler oluşturur. Böyle bir katının hacmini belirlemek için disk yöntemi kullanılabilir.Bu yöntem, katı eksene dik ince dilimlere ayırır. Her dilim, eksenden eğriye olan mesafeye eşit küçük genişlik ve yarıçapa sahip bir disk olur.Her diskin hacmi, alanının kalınlığı ile çarpılmasıyla elde edilir. Katının toplam hacmi, oluşturulan tüm dairesel disklerin hacimlerinin toplamıyla yaklaşık olarak hesaplanır.Disk sayısı arttıkça ve kalınlıkları sıfıra yaklaştıkça, toplam, verilen aralık boyunca kesit alanının belirli integralına geçer ve tam hacmi elde eder.Entegrasyon değişkeni ve yarıçapın ifadesi, eksenin yönelimine bağlıdır.

dönel cisimlerin hacimleri

Volumes of Solids of Revolution

D&#xFC;zensiz &#x15F;ekilli cisimlerin hacimleri, d&#xF6;nel cisim kavram&#x131; kullan&#x131;larak sistematik olarak belirlenebilir. Bu yakla&#x15F;&#x131;m, iki boyutlu bir d&#xFC;zlemde bir e&#x11F;ri ile tan&#x131;mlanan bir b&#xF6;lgeyle ba&#x15F;lar. S&#xF6;z konusu b&#xF6;lge, d&#xF6;nme ekseni olarak adland&#x131;r&#x131;lan sabit bir do&#x11F;ru etraf&#x131;nda d&#xF6;nd&#xFC;r&#xFC;ld&#xFC;&#x11F;&#xFC;nde, d&#xF6;nel simetriye sahip &#xFC;&#xE7; boyutlu bir cisim olu&#x15F;ur. Bu t&#xFC;r cisimler, matematiksel modelleme, fizik ve m&#xFC;hendislik uygulamalar&#x131;nda s&#x131;kl&#x131;kla g&#xF6;r&#xFC;l&#xFC;r.
D&#xF6;nd&#xFC;r&#xFC;len b&#xF6;lge do&#x11F;rudan d&#xF6;nme eksenine biti&#x15F;ik oldu&#x11F;unda, ortaya &#xE7;&#x131;kan kat&#x131; cisim, i&#xE7;i bo&#x15F;luk i&#xE7;ermeyen dairesel kesitlere sahip bir cisimdir. Bu durumda, hacmin hesaplanmas&#x131;nda disk y&#xF6;ntemi etkili bir teknik sunar. Kat&#x131; cisim, kavramsal olarak d&#xF6;nme eksenine dik olarak al&#x131;nan &#xE7;ok say&#x131;da son derece ince dilime ayr&#x131;l&#x131;r. Her bir dilim k&#xFC;&#xE7;&#xFC;k bir kal&#x131;nl&#x131;&#x11F;a sahiptir ve yar&#x131;&#xE7;ap&#x131;, eksenden ilgili noktadaki s&#x131;n&#x131;r e&#x11F;risine olan mesafeyle belirlenen bir disk olu&#x15F;turur.
Her disk, hacmi dairesel alan&#x131;n&#x131;n kal&#x131;nl&#x131;&#x11F;&#x131; ile &#xE7;arp&#x131;lmas&#x131;yla yakla&#x15F;&#x131;k olarak hesaplanabilen basit bir geometrik bir kat&#x131;y&#x131; temsil eder. Tek bir disk kat&#x131; cismin yaln&#x131;zca k&#xFC;&#xE7;&#xFC;k bir b&#xF6;l&#xFC;m&#xFC;n&#xFC; yakla&#x15F;&#x131;k olarak ifade etse de, disklerin t&#xFC;m&#xFC; birlikte ele al&#x131;nd&#x131;&#x11F;&#x131;nda cismin tamam&#x131;n&#x131; olduk&#xE7;a iyi temsil eder. T&#xFC;m disklerin hacimlerinin toplanmas&#x131;, toplam hacmin yakla&#x15F;&#x131;k de&#x11F;erini verir. Dilimler giderek inceldik&#xE7;e, bu yakla&#x15F;&#x131;m&#x131;n do&#x11F;rulu&#x11F;u artar.
Her bir diskin kal&#x131;nl&#x131;&#x11F;&#x131;n&#x131;n s&#x131;f&#x131;ra yakla&#x15F;t&#x131;&#x11F;&#x131; ve dilim say&#x131;s&#x131;n&#x131;n &#xE7;ok b&#xFC;y&#xFC;k oldu&#x11F;u s&#x131;n&#x131;rlay&#x131;c&#x131; durumda, disk hacimlerinin toplam&#x131; belirli integral arac&#x131;l&#x131;&#x11F;&#x131;yla tam bir hesaplamaya d&#xF6;n&#xFC;&#x15F;&#xFC;r. &#x130;ntegral, kat&#x131; cismin uzunlu&#x11F;u boyunca kesit alanlar&#x131;n&#x131; toplayarak d&#xF6;nel cismin tam hacmini verir. &#x130;ntegrasyon de&#x11F;i&#x15F;keninin se&#xE7;imi ve disk yar&#x131;&#xE7;ap&#x131;n&#x131; ifade eden ifade, d&#xF6;nmenin yatay ya da dikey bir eksen etraf&#x131;nda ger&#xE7;ekle&#x15F;mesine ba&#x11F;l&#x131;d&#x131;r.
Disk y&#xF6;ntemi, analizin temel ara&#xE7;lar&#x131;ndan biridir ve d&#xF6;nme yoluyla elde edilen d&#xFC;zg&#xFC;n kat&#x131; &#x15F;ekilleri i&#xE7;eren problemlerde yayg&#x131;n olarak kullan&#x131;l&#x131;r. Kavramsal yal&#x131;nl&#x131;&#x11F;&#x131; ve geometrik yorumu, d&#xF6;nel simetriye sahip fiziksel sistemlerin ve m&#xFC;hendislik bile&#x15F;enlerinin analizinde bu y&#xF6;ntemi &#xF6;zellikle de&#x11F;erli k&#x131;lar.

Dönel Cisimlerin Hacimleri

D&#xFC;zensiz &#x15F;ekilli cisimlerin hacimleri, d&#xF6;nel cisim kavram&#x131; kullan&#x131;larak sistematik olarak belirlenebilir. Bu ...

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disk method for rotated region volume

The disk method is used to find the volume of a solid formed by rotating a region around an axis. The region is first drawn in a two-dimensional plane. When that region spins around a fixed line, called the axis of revolution, it creates a three-dimensional shape with rotational symmetry. These shapes appear often in math models, physics, and engineering.

When the region touches the axis of revolution, the solid has no hollow space inside. Its cross-sections are circular, so the shape can be broken into many thin disks. Each disk is taken perpendicular to the axis of rotation. The radius of each disk comes from the distance between the axis and the curve at that point.

Each disk has a small thickness and a circular area. Its volume can be estimated by multiplying that area by the thickness. One disk gives only an approximation, but many disks together give a close model of the whole solid. As the slices become thinner and more numerous, the estimate becomes more accurate.

In the limit, this sum of disk volumes becomes an exact volume through definite integration. The integral adds the cross-sectional areas along the length of the solid. The variable of integration and the radius expression depend on whether the axis of rotation is horizontal or vertical. This method is a key idea in calculus for finding the volume of smooth, rotated shapes.

Disk Method for Rotated Region Volume

area between curves: integrating with respect to x

area-between-curves-integrating-with-respect-to-x

Vertical Slices for Area Between Curves

Area between curves can be found by using vertical slices and a definite integral. For two continuous functions on a closed interval from a to b, the region is bounded by their graphs on the sides and by the interval endpoints across the bottom and top edges of the setup. A first estimate comes from splitting the interval into many narrow vertical strips of equal width. Each strip is treated like a rectangle. Its height is the vertical difference between the two functions at a chosen point...

area between curves: integrating with respect to y

area-between-curves-integrating-with-respect-to-y

Horizontal Slices for Area Between Curves

Area between curves can be found with horizontal slices when both boundaries are written as functions of y. The left and right curves are continuous from y = c to y = d, and those lines set the vertical limits of the region. Because the curves depend on y, the area is best measured by moving across the shape with thin horizontal strips. The method uses a Riemann sum. The region is divided into many thin strips with the same height, delta y. For each strip, the width is the horizontal distance...

area between curves: problem solving

Piecewise Area Between Curves

Piecewise area between curves uses more than one integral to find the size of a region on a graph. In this example, the region is enclosed by a square root function, a reflected cube root function, and a linear function. The line intersects each of the other two curves, and those intersection points show where the boundary of the region changes. The area cannot be found with one setup over the full interval because the top and bottom curves are not the same everywhere. The region is split at...

calculation of volume of solids by integration

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Slicing Methods for Solid Volume

Slicing methods help find the volume of solids with changing cross-sections. A basic solid, like a rectangular box, is easy to measure because its base area stays the same along its height. Many real objects do not have that regular shape, so elementary geometry is not enough. The slicing method solves this by breaking a solid into many very thin slices. These slices are taken perpendicular to a chosen axis, often the x-axis. Each slice has a small thickness, written as delta x, and a...

volumes of solids of revolution

The disk method is used to find the volume of a solid formed by rotating a region around an axis. The region is first drawn in a two-dimensional plane. When that region spins around a fixed line, called the axis of revolution, it creates a three-dimensional shape with rotational symmetry. These shapes appear often in math models, physics, and engineering. When the region touches the axis of revolution, the solid has no hollow space inside. Its cross-sections are circular, so the shape can be...

finding volume using cross-sectional area

finding-volume-using-cross-sectional-area

Tetrahedron Volume from Triangle Slices

Tetrahedron volume can be found by adding the areas of triangular slices. This method uses cross-sectional area, which is the area of a flat slice taken through a 3D solid. It is useful when a solid changes shape in a regular way and a simple geometric formula is not available. A tetrahedron is a good example. It has a height h and an equilateral triangle base with side length a. The coordinate system places the origin at the top vertex, and the y-axis goes downward along the height to the...

problem solving: volume

Finding the Volume of a Jet Fuel Tank

The volume of a jet fuel tank can be found with solids of revolution. The tank is modeled by rotating a two-dimensional region around the x-axis. This creates a three-dimensional shape that is symmetric about the axis of rotation. The region runs from 0 to 2 meters, so the tank’s shape can be measured along that length. Because the boundary curve touches the axis, the disk method works well here. The disk method breaks the solid into many very thin circular slices that are perpendicular to the...

average value of a function

Estimating Function Behavior with Average Value

The average value of a function gives one number that summarizes how a quantity changes over a closed interval. It can also be viewed as the height of a rectangle whose area matches the net area under the curve. That net area includes both positive and negative parts of the function, so the result reflects the function’s overall behavior. A greenhouse temperature over a 24-hour period is a practical example of this idea. The temperature may rise and fall throughout the day, but it is still...

key terms and concepts

Learn how the disk method finds the volume of a solid of revolution by slicing a rotated region into thin disks and using definite integration.

Disk Method for Rotated Region Volume | 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

Disk Method for Rotated Region Volume

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