determine volume of region from description of views

[Smily]

can someone help me, please? :roll:

A region in space, when viewed from three different views, looks like a circle, a square, and a triangle. Describe this object. Use multiple integration to determine its volume, in terms of the diameter a

Thank you!

 

[soroban]

Re: determine its volume

Hello, Smily!

A region in space, when viewed from 3 different views,
looks like a circle, a square, and a triangle.
Describe this object.
Use multiple integration to determine its volume, in terms of the diameter \(\displaystyle a\)

We have a right circular cylinder.
The base is a circle of diameter \(\displaystyle a\); its height is \(\displaystyle a.\)
. . Viewed from above, it looks like a circle.
. . Viewed from the side, it looks like a square.

Looking from the "front", slice the cylinder so it looks like an isosceles triangle.

      ......*......
      :    /:\    :
      :   / : \   :
      :  /  :a \  :
      : /   :   \ :
      :/    :    \:
      *-----------*
              a/2

Place the object on the xy-plane, centered at the origin.
Due to its symmetry, we can find the volume in the first octant and multiply by 4.

The volume is above \(\displaystyle z\,=\,0\) and below the plane \(\displaystyle z \:=\:a\,-\,2y\)

The base is the circle: \(\displaystyle \,x^2\,+\,y^2\:=\:\left(\frac{a}{2}\right)^2\)
So \(\displaystyle y\) ranges from \(\displaystyle y\,=\,0\) to \(\displaystyle y\:=\:\frac{1}{2}\sqrt{a^2\,-\,4x^2}\)

Finally, \(\displaystyle x\) ranges from \(\displaystyle x\,=\,0\) to \(\displaystyle x\,=\,\frac{a}{2}\)


. . . . . . . . . . . . . . . . . . . . . . . . . \(\displaystyle _{\frac{a}{2}}\;\;_{\frac{1}{2}\sqrt{a^2-4x^2}}\;\,_{a-2y}\)
I think the volume is: \(\displaystyle \L\:V \;= \;4\int_0\int_0\;\;\;\;\;\int_0\;\;dz\,dy\,dz\)

. . but don't quote me . . .