![](https://wkrtcs.bdimg.com/rtcs/image?w=542.67&md5sum=cd04dfb2cd751e473417e5efa93bbc12&sign=8724c996a3&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=16&ipr={"t":"img","r":"r_0","w":"542.67","h":"320.00","dataType":"gif","c":"word/media/image1.gif"})
一个横放的椭圆截面的圆柱容器，里面有液体，液体高度如果为任意值H，怎么算其中液体体积？？容器外部尺寸都可以测量![](https://wkrtcs.bdimg.com/rtcs/image?w=47.00&md5sum=cd04dfb2cd751e473417e5efa93bbc12&sign=8724c996a3&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=16&ipr={"t":"img","w":"47.00","h":"21.00","dataType":"gif","c":"word/media/image2.gif"})
设椭圆方程为：![](https://wkrtcs.bdimg.com/rtcs/image?w=76.00&md5sum=cd04dfb2cd751e473417e5efa93bbc12&sign=8724c996a3&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=16&ipr={"t":"img","w":"76.00","h":"44.00","dataType":"gif","c":"word/media/image10.gif"})
（1）当![](https://wkrtcs.bdimg.com/rtcs/image?w=42.67&md5sum=cd04dfb2cd751e473417e5efa93bbc12&sign=8724c996a3&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=16&ipr={"t":"img","w":"42.67","h":"18.67","dataType":"gif","c":"word/media/image11.gif"})
时，只算第一象限的阴影部分面积，然后根据对称性可得体积
则阴影部分面积为（其中![](https://wkrtcs.bdimg.com/rtcs/image?w=65.33&md5sum=cd04dfb2cd751e473417e5efa93bbc12&sign=8724c996a3&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=16&ipr={"t":"img","w":"65.33","h":"18.67","dataType":"gif","c":"word/media/image12.gif"})
）：
![](https://wkrtcs.bdimg.com/rtcs/image?w=161.33&md5sum=cd04dfb2cd751e473417e5efa93bbc12&sign=8724c996a3&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=16&ipr={"t":"img","w":"161.33","h":"48.00","dataType":"gif","c":"word/media/image13.gif"})
令![](https://wkrtcs.bdimg.com/rtcs/image?w=72.00&md5sum=cd04dfb2cd751e473417e5efa93bbc12&sign=8724c996a3&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=16&ipr={"t":"img","w":"72.00","h":"21.33","dataType":"gif","c":"word/media/image14.gif"})
，可得
![](https://wkrtcs.bdimg.com/rtcs/image?w=232.00&md5sum=cd04dfb2cd751e473417e5efa93bbc12&sign=8724c996a3&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=16&ipr={"t":"img","w":"232.00","h":"41.33","dataType":"gif","c":"word/media/image15.gif"})
![](https://wkrtcs.bdimg.com/rtcs/image?w=185.33&md5sum=cd04dfb2cd751e473417e5efa93bbc12&sign=8724c996a3&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=16&ipr={"t":"img","w":"185.33","h":"41.33","dataType":"gif","c":"word/media/image16.gif"})
![](https://wkrtcs.bdimg.com/rtcs/image?w=321.33&md5sum=cd04dfb2cd751e473417e5efa93bbc12&sign=8724c996a3&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=16&ipr={"t":"img","w":"321.33","h":"133.33","dataType":"gif","c":"word/media/image17.gif"})
此时液体体积为：
![](https://wkrtcs.bdimg.com/rtcs/image?w=116.00&md5sum=cd04dfb2cd751e473417e5efa93bbc12&sign=8724c996a3&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=16&ipr={"t":"img","w":"116.00","h":"41.33","dataType":"gif","c":"word/media/image18.gif"})
其中![](https://wkrtcs.bdimg.com/rtcs/image?w=17.33&md5sum=cd04dfb2cd751e473417e5efa93bbc12&sign=8724c996a3&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=16&ipr={"t":"img","w":"17.33","h":"14.67","dataType":"gif","c":"word/media/image19.gif"})
表示容器的长度
（2）当![](https://wkrtcs.bdimg.com/rtcs/image?w=42.67&md5sum=cd04dfb2cd751e473417e5efa93bbc12&sign=8724c996a3&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=16&ipr={"t":"img","w":"42.67","h":"18.67","dataType":"gif","c":"word/media/image20.gif"})
时，只算第一象限的阴影部分面积，然后根据对称性可得体积
![](https://wkrtcs.bdimg.com/rtcs/image?w=62.67&md5sum=cd04dfb2cd751e473417e5efa93bbc12&sign=8724c996a3&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=16&ipr={"t":"img","w":"62.67","h":"6.67","dataType":"gif","c":"word/media/image21.gif"})
图中两个阴影部分面积相等，因此需算上面那部分的面积即可得知下面那部分的面积。
此时令![](https://wkrtcs.bdimg.com/rtcs/image?w=65.33&md5sum=cd04dfb2cd751e473417e5efa93bbc12&sign=8724c996a3&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=16&ipr={"t":"img","w":"65.33","h":"18.67","dataType":"gif","c":"word/media/image32.gif"})
，
此部分面积为：
![](https://wkrtcs.bdimg.com/rtcs/image?w=124.00&md5sum=cd04dfb2cd751e473417e5efa93bbc12&sign=8724c996a3&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=16&ipr={"t":"img","w":"124.00","h":"48.00","dataType":"gif","c":"word/media/image33.gif"})
![](https://wkrtcs.bdimg.com/rtcs/image?w=224.00&md5sum=cd04dfb2cd751e473417e5efa93bbc12&sign=8724c996a3&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=16&ipr={"t":"img","w":"224.00","h":"48.00","dataType":"gif","c":"word/media/image34.gif"})
此时液体体积为：
![](https://wkrtcs.bdimg.com/rtcs/image?w=65.33&md5sum=cd04dfb2cd751e473417e5efa93bbc12&sign=8724c996a3&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=16&ipr={"t":"img","w":"65.33","h":"24.00","dataType":"gif","c":"word/media/image35.gif"})
其中表示容器的长度
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