Geometry and Calculus provide the mathematical foundations for understanding the volume of a cone. To derive the formula for the volume of a cone (1/3)πr²h, we can use the method of calculating the volume of a pyramid and then take the limit as the number of slices approaches infinity. Imagine a cone with a base radius r and a height h. Divide the cone into several frustums of cones, or "cone slices," along the height h of the cone. Each frustum has a height h', and the base radii change from r1 to r2. The volume of each frustum of a cone can be calculated using the formula for the volume of a pyramid: (1/3)Ah, where A represents the area of the base and h is the height of the pyramid or frustum. In our case, the base of each frustum is a circle, so the area A of the base can be calculated as πr² for a full cone. For a frustum, the area A can be calculated as π[(r1 + r2)/2]². Now, sum up the volumes of these frustums from the apex to the base of the cone. As the number of frustums approaches infinity, the summation transforms into an integral. The definite integral of the function representing the volume of these frustums is evaluated from 0 to h. The integral is then (1/3)πr²h + C, where C is the constant of integration. Since the volume of a cone at the apex is zero, C is equal to zero. Thus, the volume of a cone is (1/3)πr²h. In summary, the volume of a cone is one third of the volume of a cylinder with the same base and height due to the shape's tapering geometry. The formula can be derived from the summation of frustums of cones and the application of integral calculus.
@@bprptw The equation of a straight line is y=mx+c , where m is the slope and c is a constant. In this case, m is h/r and c is a constant -0 when y=0 and x =0 , c=0 as you can see the line passes through x=0 and y=0.
We can derive the formula for the volume of a cone using integration. Here is a detailed explanation: Consider a right circular cone with a base radius r and a height h. Place the cone on its side with the apex (or vertex) pointing up and the base lying flat on the xy-plane. Now, slice the cone parallel to the xy-plane, dividing it into infinitely many circular disks (washers) of thickness dz. The thickness dz of each disk is an infinitesimally small change in the z-coordinate along the height (altitude) of the cone. As z changes from 0 to h, the radius of each disk changes from 0 to r. The area of each disk (washer) is given by the formula for the area of a circle, A = πr². The radius r of each disk can be expressed as a function of z, where r = (z/h)r. The volume of each disk can be calculated as its area (A) times its thickness (dz): dV = A dz = π[(z/h)r]²dz. Now, to find the total volume of the cone, we need to sum up the volumes of all these infinitesimally small disks (washers) along the height of the cone. This summation can be expressed as an integral, and we will integrate the volume of each disk from z = 0 to z = h: V = ∫[0, h] π[(z/h)r]²dz Integrating this expression, we get: V = (1/3)πr²h Thus, the volume of a cone is (1/3)πr²h, which can be derived using integral calculus. In summary, the volume of a cone is (1/3)πr²h because, as we slice the cone infinitely thin along its height and calculate the volume of each infinitesimally small disk (washer), we can sum the volumes of all these disks using integration. This yields the formula for the volume of a cone.
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太巧，剛剛在自己頻道寫完球體體積的文，就出現錐體體積的影片了。
話說回來，其實在積分之前就有各種方式計算體積問題， 而且更接近其內涵，積分只是純操作計算的大統整。
我在中學時，即使已經親手推了好幾遍 體積公式 了，也知道阿基米德會如何證明錐體的 1/3 ，但我那時依然不知道積分是什麼東西。
非常棒。小学时候老师只将到装水的那个步骤就结束了，这下我算是明白这中间的数学含义了
小學老師跟你講微積分你也聽不懂
THX to illustrate this principle and theorm again... I just remember and easily remember wrong!!
我最近剛好也在找椎體體積問題，雖然不會積分，但看到有個頻道用邊長為r的正立方體切六分（以質心做六四角錐的頂點，接著1/6r³=1/3•1/2r•r² （1/2r為四角錐柱高）
以此推知椎體為1/3•底面積•高
只能算推論，不能算嚴謹的證明
不仅仅是圆锥型，任何锥形都是1/3
平時看都是英文版今天終於看到了中文版. VCD 跟 CD 中的CD 來自 Compact Disc. 但是 A:\ drive (1.44 英吋 floppy disk drive) 用的 floppy disk 也有時候叫做 diskette 因為比 B:\ drive 較小, B:\ drive (5.25英吋)的 時候是用 disk 因為來自 Microsoft 公司用語的 C:\ drive (hard disk drive, 簡稱 hard drive) 而中文翻譯成 “硬盤” 之類的，等等。。。。。。才用 disk.
其实更一般的做法可以直接证明任意“锥体”的体积都是1/3 Sh，其中S是底面积，h是高。
所谓“锥体”就是体面随着高增加等比例增加的立体形状。
所以S 正比于 h^2 (相似图形的面积正比于线度的平方）
可以写成S = c h^2
简单积分\int_0^h S dh = 1/3 c h^3 = 1/3 Sh
你這做法的前提是
必須接受“底面積會正比於高度平方”這件事，才能夠算式這麼簡單~
因此雖然你這方法看似“更簡易”，但卻沒有“更基礎”~
@@mujium 这个从物理和古典数学的角度就是更基础。因为任何形状缩放时面积平方变化
@@howareyou4400所以從純數學的角度，你這裡得拉兩條引理：
1. 椎體截面面積必然相似
2. 平面上相似多邊形面積比例為邊長比例的平方
@@吳小天-v8z
1. 椎体可以缩放出大小椎体，侧面始终重合
2. 相似的平面图形（不一定是多边形）面积比例为相似比的平方，相似比定义为缩放x倍之后重合。（这一条可以当成基本定理甚至公理的）
这个确实不是现代数学的超严谨写法，但是是现代物理学以及近代数学认可的开式。
我们普通人学的数学其实也就到近代数学的程度。
@@mujium那是事實，你接不接受不會影響事實? 我不知道你在講什麼耶？噗
中學時學到的證明:
利用Cavalieri's principle: 兩個高度一樣, 任意平行底面的截面面積積也都一樣的物體體積相等
可以得到
1. 所有底面積和高相同的錐體不論歪斜的程度和底面的形狀為何體積都相等
2. 設三角柱頂面為 三角形ABC 底面為三角形A'B'C'
則三角錐 ABCB' 體積 = 三角錐 A'B'C'C 體積 = 三角錐 A'C'CB' 體積
因此三角錐體積為三角柱的 1/3
=> 任意錐體的體積為同底等高柱體的 1/3
你的最後一步，需要先證明n個底角的n角錐是n角柱體積的1/3，然後證明n趨近無限大時仍然收斂，從純數學證明來講，反而更困難😂
根據Cavalieris principle 錐體體積跟底面形狀無關，所以不需要證明n邊形
❤
@@YJChu👍
微積分還沒發現前應該有人證明過嗎?
同問
中國古代能求出四角錐的體積是等高長方體的1/3，然後根據祖暅原理，兩個錐體每一處高度對應的截面積相同的話總體積也會相同。雖然不叫微積分，但其實就有隱藏那個概念
應該是由角錐的公式歸納出的圓錐公式
然後角錐公式的話
想像一個高為底邊長一半的正方錐
我們可以不切割的情況下把六個這樣的錐體剛好填滿一個邊長等於該正方錐底邊的立方體
即 6 x 正方錐體積 = 立方體體積
而因為正方錐的高只有立方體邊長的一半，把上述等式化簡後可得出 正方錐體積 = 1/3 底面積 x 高
啊，留完言才發現你的留言。
答案是當然有，另一位大大講的祖暅(ㄒㄩㄢ)原理記憶中是南北朝時期的數學家，祖沖之的兒子 。 不過再更早的阿基米德就有方法應付這種體積問題了，有力學的方法也有數學的方法。
印象用拆分
感謝曹老師講得淺顯易懂，我不是理組也不是數學系的，但是我沒有很怕數學，反正就加減聽這樣。😂😂😂
5分40秒時突然說出三分之一這個關鍵數字，作為聽講者，我實在不知道三分之一是怎麼冒出來的。
積分心算 太基本了所以不解釋過程
積分時，本來二次的要加1變成三次方，前面的係數就是1/3,真不懂只能去看積分，稍微入門就有提到了
y3次方微分=3y平方，所以y平方積分=(y3次方)/3，
以錐體頂點為0，對每個截面求截面積
設截面高為t，將t以h的倍數形式表示
可以得到截面積 = πr²·t²/h²
接著對t積分，得到
體積 = 1/3·πr²·t³/h²
t=h時，截面積為πr²，體積為1/3·πr²h
t=1/2h時，截面積為1/4πr²，體積為1/24·πr²h
而這個得出的 ＂1/3＂，在現實世界中有一定意義，
這好像是我之前看伊莉亞動畫
他們在教這個單元
結果美遊上台用積分算出來
的那時候學會的
Geometry and Calculus provide the mathematical foundations for understanding the volume of a cone.
To derive the formula for the volume of a cone (1/3)πr²h, we can use the method of calculating the volume of a pyramid and then take the limit as the number of slices approaches infinity.
Imagine a cone with a base radius r and a height h.
Divide the cone into several frustums of cones, or "cone slices," along the height h of the cone. Each frustum has a height h', and the base radii change from r1 to r2.
The volume of each frustum of a cone can be calculated using the formula for the volume of a pyramid: (1/3)Ah, where A represents the area of the base and h is the height of the pyramid or frustum.
In our case, the base of each frustum is a circle, so the area A of the base can be calculated as πr² for a full cone.
For a frustum, the area A can be calculated as π[(r1 + r2)/2]².
Now, sum up the volumes of these frustums from the apex to the base of the cone.
As the number of frustums approaches infinity, the summation transforms into an integral.
The definite integral of the function representing the volume of these frustums is evaluated from 0 to h.
The integral is then (1/3)πr²h + C, where C is the constant of integration.
Since the volume of a cone at the apex is zero, C is equal to zero.
Thus, the volume of a cone is (1/3)πr²h.
In summary, the volume of a cone is one third of the volume of a cylinder with the same base and height due to the shape's tapering geometry. The formula can be derived from the summation of frustums of cones and the application of integral calculus.
1:45
y=(h/r)*x
請問這是在寫什麼東東？
那條線的方程式, 從(0,0) 到 (r, h)
十多年前我學習微分，積分，歸納的時候，我的老師也是一直跟我說“總之就是會變成這個樣子”，結果十多年後的現在我還是不懂怎樣算
是用斜率和某x的值來找出對應的y值吧
我想這應該是點斜式吧 (y - 0) = h/r * (x - 0)
@@bprptw The equation of a straight line is y=mx+c , where m is the slope and c is a constant. In this case, m is h/r and c is a constant -0
when y=0 and x =0 , c=0 as you can see the line passes through x=0 and y=0.
高三學基礎微積分的時候 參考書就有一系列體積證明的題目 那時才解決自國中以來的疑惑
不過我不是用影片的切法 是用中空圓柱去切的
要知道一個柱體是由三個錐體合成的很簡單 如果是要用數學計算來證明就不是一般人做得到的甚至是聽不懂啦
有字幕會更好 ...這樣不開聲音沒戴耳機也能看
4天后高等微积分大考～
我現在也正在學這條公式，就是沒想過原來背後這麼複雜😮😮
讚
哎呀呀！😮
希望有一些黎曼和的證明影片～
這裡有
ruclips.net/video/Cv4ZjLfkEpo/видео.html
也可以去看我100題極限
We can derive the formula for the volume of a cone using integration. Here is a detailed explanation:
Consider a right circular cone with a base radius r and a height h.
Place the cone on its side with the apex (or vertex) pointing up and the base lying flat on the xy-plane.
Now, slice the cone parallel to the xy-plane, dividing it into infinitely many circular disks (washers) of thickness dz.
The thickness dz of each disk is an infinitesimally small change in the z-coordinate along the height (altitude) of the cone.
As z changes from 0 to h, the radius of each disk changes from 0 to r.
The area of each disk (washer) is given by the formula for the area of a circle, A = πr².
The radius r of each disk can be expressed as a function of z, where r = (z/h)r.
The volume of each disk can be calculated as its area (A) times its thickness (dz): dV = A dz = π[(z/h)r]²dz.
Now, to find the total volume of the cone, we need to sum up the volumes of all these infinitesimally small disks (washers) along the height of the cone.
This summation can be expressed as an integral, and we will integrate the volume of each disk from z = 0 to z = h:
V = ∫[0, h] π[(z/h)r]²dz
Integrating this expression, we get:
V = (1/3)πr²h
Thus, the volume of a cone is (1/3)πr²h, which can be derived using integral calculus.
In summary, the volume of a cone is (1/3)πr²h because, as we slice the cone infinitely thin along its height and calculate the volume of each infinitesimally small disk (washer), we can sum the volumes of all these disks using integration. This yields the formula for the volume of a cone.
我一直很對三角函數有一個疑問，希望可以有人解答一下
在我的角度，三角函數跟f(x)是完全一樣的東西，代入x，得出的結果便是邊與邊的比例
然而，這個結果是從何而來的呢？他應該跟一個一般的f(x)一樣有一條方程式，在而透過代入x進去，就能夠算出邊與邊之間的比例
我考慮過會否是因為如同標準正態函數分佈的公式那樣的積分實在太難，因此不教，導致我找不到
然而我即使在網路上尋找，也只能找到其泰勒展開，而非原本的公式，我不認為泰勒展開的公式便是他的表達式，那對我而言只是一個臨摹的手段，正如同我可以表達f(x)=e^x=1+x+x^2/2+…，無論我把x代入後兩者哪一個得出的結果都是一樣的（假如取無窮項）
三角函數不應該也有其真正的表達式嗎？如果有？那到底是什麼樣的，我真的困惑好久了qwq
三角函數是超越函數，無法被多項式函數表示，只能逼近（像泰勒展開就是）
請問老師～ 如果反過來求一個曲面柱體 斜面不是一次式（y=r/h x） 而是某個二次式（y=a x^2 +bx+c） 使得旋轉體積變成 2/3 pi r^2 h 有辦法求出a、b、c與r、h的關係嗎？謝謝老師～～
运用旋转体体积公式V=π∫[a,b]f(x)^2dx，把y=ax^2 +bx+c带入公式可以求，但是需要你提供定积分函数上下限，你所提供的h只能知道上下限的差
微積分大概學分拿到就還老師了XD
以前经常追，后来就没再追了，今天突然发现成中文了，竟然有中文版😮
😆
its like i can understand him, but no
There’s an English version of this proof on my main channel.
其實最原始的定理最難證明，為甚麼圓周長是2πr，相信一堆人不知道怎麼證明
國小課本上我記得有
這是定義，不用證明
π的定义就是圆周长和两倍半径的比值😂
讲的不好，逻辑不完备
哈..pi*r^2 也是用微積分算出來的
牛吨，真特么吨。到底是牛顿还是莱布尼茨发明的这玩意?微积分?