1 . 祖暅是我国南北朝时期的数学家,著作《缀术》上论及多面体的体积:缘幂势既同,则积不容异——这就是祖暅原理.用现代语言可描述为:夹在两个平行平面之间的两个几何体,被平行于这个两个平面的任意平面所截,如果截得的两个截面的面积总相等,那么这两个几何体的体积相等.在棱长为2的正方体
中,
是
上一点,
于点
,
,点
绕
旋转一周所得圆的面积为_________ (用
表示);将空间四边形
绕
旋转一周所得几何体的体积为_________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
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390次组卷
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4卷引用:上海市宜川中学2024届高三下学期2月开学考试数学试题
2 . 在
平面上,将一段圆弧
:
(
)和一段椭圆弧
:
(
)围成的封闭图形记为
,如图中阴影部分所示,
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/7/bb28ce6e-097b-49b4-a411-0da00b1548a1.png?resizew=135)
记
绕
轴旋转一周而成的封闭几何体为
,过
(
)作
的水平截面,利用祖暅原理和一个球,得出旋转体
的体积值为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
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![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/7/bb28ce6e-097b-49b4-a411-0da00b1548a1.png?resizew=135)
记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
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3 . 如图所示,已知一个半径为2的半圆面剪去了一个等腰三角形
,将剩余部分绕着直径
所在直线旋转一周得到一个几何体,其中点
为半圆弧的中点,该几何体的体积为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
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![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/13/1ec112e5-7e60-4037-89cd-40a71f53f393.png?resizew=82)
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解题方法
4 . 已知双曲线方程
,直线
,
在第一象限内与双曲线及渐近线围成的图形绕
轴旋转一周所得几何体的体积为______ .(提示:利用祖暅原理)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e1fa37c4c826b5dcfebe86ab6177906.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2a7df955fc17e92fd86302f8c34664a.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
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5 . 已知圆锥的母线与底面所成的角为
,体积为
,则圆锥的底面半径为___________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac1a63ab608517bb10aa036783dfb51f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cf1f865bafd4a820406d336d99f8091.png)
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6 . 在
平面上,将两个函数
和
、两条直线
和
围成的封闭图形记为
,如图所示,记
绕
轴旋转一周而成的几何体为
,则
的体积值________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78b91d90e35deba1cdc76b3247d8d909.png)
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![](https://img.xkw.com/dksih/QBM/editorImg/2023/7/11/81df9384-c6f4-4fd4-bc9c-cf0544733343.png?resizew=205)
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7 . 如图所示(单位:cm),直角梯形ABCD挖去半径为2的四分之一圆,则图中阴影部分绕AB旋转一周所形成的几何体的体积为__ .
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/26/f8444c38-a828-4a7c-8d3e-cf98ca9af57f.png?resizew=155)
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8 . 已知四棱锥
的底面是边长为
的正方形,侧棱长均为
.若点
在圆柱的一个底面圆周上,点P在圆柱的另一个底面内,则该圆柱的体积为__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf298f00799cbf34b4db26f5f63af92f.png)
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4卷引用:上海市嘉定区2023届高三二模数学试题
上海市嘉定区2023届高三二模数学试题(已下线)专题07 空间向量与立体几何上海市宜川中学2023-2024学年高三上学期期中考试数学试题(已下线)第四章 立体几何解题通法 专题五 平移变换法 微点2 平移变换法(二)【培优版】
9 . 记双曲线的一支
,曲线
和直线
所围成的封闭图形为
,将
绕着直线
旋转一周所形成的几何体记作
,试利用祖暅原理、一个圆柱和一个圆锥,得出
的体积值为___________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60a3cb98cd35fcda6c29629a0b0b7fa1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34715101c66fa12ce6baf0a9c53f1672.png)
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解题方法
10 .
中,
,过点A的直线
在平面
上,且
在直线
的同一侧,将
绕直线
旋转一周所得的几何体的体积的最大值为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7061007045e5440a9f021a6c12fc615.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
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