Expert: Scotto - 1/7/2011

Question
QUESTION: 1.how to derive the formulas of volumes of cylinder,cone&sphere??
2.how to derive the formulas of areas of circle & triangle??

ANSWER: 1. The volume a cylinder is the area of the base times the height.
The area of the base is the area of a circle.

The volumn of a cone is one third the area of the base times the height.

The volume of a sphere is (4/3)(pi)r^3, where r is the radius.
The number pi is 3.1416 (when rounded to four places; it is really infinite).

2. The area of a circle is pi*r^2, where r is the radius.

The area of a triangle is bh/2 where b is the length of the base and
h is the perpendicular height.


---------- FOLLOW-UP ----------

QUESTION: i want to ask that how to derive the formulas of volumes of cylinder & other solids using calculus

Answer
Picture the cyllinder sideways with the center of each circle on the x axis.
Since the area of a circle is pi*r², integrate this constant from 0 to the height of the cyllinder.  The answer is just pi*r²*x from 0 to h, where h is the height.
This comes out to be pi*r²*h,

For a rectangular solid, it could be done with a triple integral.
The volume equation to integrate is 1, with the height, width, and length as W, H, and L.
The 1st integral would be from 0 to W, the 2nd integral would be from 0 to H,
and the 3rd integral would be from 0 to L.

For a cyllinder, it can also be converted to cyllindrical coordinates,
where dx dy is replaced with with r dr dA, where r is the radius and A is the angle.

For a sphere, dx dy dz can be replace with r²sinA dA dB where the angles are A and B.
Once this is done, A and B both go from 0 to 2*pi and r goes from 0 to the radius.

The conversion depends a lot on what type of volume is being found.