1 . 我们要计算由抛物线
、
轴以及直线
所围成的曲边区域的面积
,可用
轴上的分点0、
、
、…、
、1将区间
分成
个小区间,在每一个小区间上作一个小矩形,使得每个矩形的左上端点都在抛物线
上,这么矩形的高分别为0、
、
、…、
、1,矩形的底边长都是
,设所有这些矩形面积的总和为
,就无限趋近于
,即
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/29/f225ce7f-2b36-4860-9093-546a431495be.png?resizew=171)
(1)求数列
的通项公式,并求出已知
;(可以利用公式
)
(2)利用上述方法,探求由函数
、
轴、
轴以及直线
和所围成的区域的面积
.(可以利用公式:
)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/344ccbf79da6ad7e3709d6fa72efb756.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b384412acba251d87902ab928902f16.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee6a2eba56d4f2d1670b0256b8d86b92.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c350e626893ba12c028b6c8bc4ea8d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cc7040ad5dd0ebafac0e9f23bd54812.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e11f4ca0e7ace69f92130d0525bcdb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/344ccbf79da6ad7e3709d6fa72efb756.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b176553f857ff6e6641d79c9e83be10.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ef7d1d8923879a6fdaf839d87aafeff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/adf343294853aef9035ed93efb09ec2a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee6a2eba56d4f2d1670b0256b8d86b92.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a12c3439a5339dd2ae4c91e52466f3c.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/29/f225ce7f-2b36-4860-9093-546a431495be.png?resizew=171)
(1)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76a16dcc0f10910231b301311a3f4f02.png)
(2)利用上述方法,探求由函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f2eff609c6043c2a89a6dd163fe2244.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b384412acba251d87902ab928902f16.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e28ccf6acba4afeb5983cc1bc575d13b.png)
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2020高三·全国·专题练习
2 . 计算下列定积分:
(1)
;(2)
;(3)
.
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed63e1e4cb496cec4418a46d4dbb6332.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a0b2ddcce42034c7943e1885ffff8dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4963bb3fe5f5a0f666bb81d34b17feb.png)
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3 . 如图:已知
通过点(1,2),与
有一个交点横坐标为
,且
.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/7/0c6addd3-0a6b-478f-bba2-ebf1ebce798c.png?resizew=255)
(1)求
与
所围的面积
与
的函数关系;
(2)当
为何值时,
取得最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53be61cd85ec86aabd164cae0265246b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef7a6d0032f9161eccda678c0d1d9d79.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13727f8b48c102f85634d46413addda6.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/7/0c6addd3-0a6b-478f-bba2-ebf1ebce798c.png?resizew=255)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53be61cd85ec86aabd164cae0265246b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef7a6d0032f9161eccda678c0d1d9d79.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/632244ea6931507f8656e1cc3437d392.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
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4 . 利用定积分的几何意义求
,其中
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7fd3da06fe0621235a844ef6e8687223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca66bbedc244e3cc55d5b4203f75607e.png)
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