Question

how does cavalieris principle help explain how to find the volume of a cylinder? a cylinder has a circular base and you can think of it as circular disks piled on top of one another to form the height. this height changes if the disks are not piled straight up, so it only applies to right cylinders. a cylinder has a circular base and you can think of it as circular disks piled on top of one another, but each disk is a different height, so you have to know the height of each one. a cylinder has a circular base and you can think of it as many circular disks. if you spread them out and find their surface area, you will then know the volume of the cylinder. a cylinder has a circular base and you can think of it as circular disks piled on top of one another to form the height. this makes the volume equal to the area of the base times the height.

Explanation:

Cavalieri's Principle states that if two solids have the same height and the same cross - sectional area at every level, then they have the same volume. A cylinder can be thought of as a stack of circular disks (with the same cross - sectional area, which is the area of the circular base) piled on top of one another to form the height. The volume of each disk is approximately \(A\times\Delta h\) (where \(A\) is the area of the base and \(\Delta h\) is a small height increment). Summing up these (using the concept similar to what Cavalieri's Principle implies, as the cross - sectional area \(A\) is constant), the volume of the cylinder \(V = A\times h\) (where \(A=\pi r^{2}\) for a circular base and \(h\) is the height of the cylinder).

The first option is wrong because Cavalieri's Principle applies to both right and oblique cylinders (as long as the cross - sectional area is constant at every height). The second option is wrong because in the context of Cavalieri's Principle for a cylinder, the disks have the same height increment (when we use the limit process for volume calculation). The third option is wrong because surface area of the disks is not used to find the volume of the cylinder.

Answer:

A cylinder has a circular base and you can think of it as circular disks piled on top of one another to form the height. This makes the volume equal to the area of the base times the height.