Volume difference between cylinder and a cube?

[pipc]

A satellite is created as the intersection between two cylinders with the same radius and perpendicular symmetry axis.
Is is transferred from the factory to the launching pad in a cube shaped box.

What part of the Volume of the cubic box is unused?

 

[Deleted member 4993]

A satellite is created as the intersection between two cylinders with the same radius and perpendicular symmetry axis.
Is is transferred from the factory to the launching pad in a cube shaped box.

What part of the Volume of the cubic box is unused?

Have you taken Calculus?

 

[galactus]

When we intersect two cylinders of the same radius, the cross sections are squares.

With axes through the center of one of the cylinders and perp. to the axis of the cylinder, with y

directed upwards, the squares have side length at level y of $\displaystyle 2\sqrt{r^{2}-y^{2}}$.

The area of the cross section(area of square) of the intersection at level y is then:

$\displaystyle A(y)=(2\sqrt{r^{2}-y^{2}})^{2}=4(r^{2}-y^{2})$

This is what is to be integrated: $\displaystyle 4\int_{-r}^{r}(r^{2}-y^{2})dy$

Now, what is the volume of the cube the satellite is in?. Find it, then subtract the two volumes.

What remains is the unused portion.

Here is a diagram to help picture what I am trying to describe.

The gray square is a cross section looking down from the top. The circle is looking in through a cylinder. The chord through the circle is a side of a square that makes a cross section.