大一高数,求围成的体积,这儿两天正在用极坐标求
来源:学生作业帮 编辑:搜搜做题作业网作业帮 分类:数学作业 时间:2024/08/04 03:21:45
大一高数,求围成的体积,这儿两天正在用极坐标求
![](http://img.wesiedu.com/upload/8/14/81439b0880e8055d4cfa0da628fd2b7e.jpg)
![](http://img.wesiedu.com/upload/8/14/81439b0880e8055d4cfa0da628fd2b7e.jpg)
![大一高数,求围成的体积,这儿两天正在用极坐标求](/uploads/image/z/11568687-15-7.jpg?t=%E5%A4%A7%E4%B8%80%E9%AB%98%E6%95%B0%2C%E6%B1%82%E5%9B%B4%E6%88%90%E7%9A%84%E4%BD%93%E7%A7%AF%2C%E8%BF%99%E5%84%BF%E4%B8%A4%E5%A4%A9%E6%AD%A3%E5%9C%A8%E7%94%A8%E6%9E%81%E5%9D%90%E6%A0%87%E6%B1%82)
所围成的体积=∫∫(∫dz)dxdy
(S是所围成的体积在xoy平面上的投影区域:4x^2+y^2≤1)
=∫∫[(1-x^2)-(3x^2+y^2)]dxdy
=∫∫[(1-4x^2-y^2)]dxdy
=∫dθ∫(1-r^2)(r/2)dr (作变换:x=(r/2)cosθ,y=rsinθ)
=π∫(r-r^3)dr
=π(1/2-1/4)
=π/4.
(S是所围成的体积在xoy平面上的投影区域:4x^2+y^2≤1)
=∫∫[(1-x^2)-(3x^2+y^2)]dxdy
=∫∫[(1-4x^2-y^2)]dxdy
=∫dθ∫(1-r^2)(r/2)dr (作变换:x=(r/2)cosθ,y=rsinθ)
=π∫(r-r^3)dr
=π(1/2-1/4)
=π/4.