Posted by **matt** on Saturday, January 31, 2009 at 4:53pm.

Use integrals to prove that the volume of a sphere of radius

R is equal to (4/3)(pi)R^3

- calculus -
**Damon**, Saturday, January 31, 2009 at 5:06pm
well the really easy way is to assume that you know the area of a sphere is 4 pi r^2

integral dr (4 pi r^2)

= 4 pi r^3 / 3 = (4/3) pi r^3

- calculus -
**Damon**, Saturday, January 31, 2009 at 5:17pm
Otherwise radius of plane parallel to x,y axis as a function of height above the center plane is

r = R cos theta

area of circular plane at height z= R sin theta = pi r^2 = pi R^2 cos^2 theta

so hemisphere above axis

integral dz pi R^2 cos^2 theta

but dz = d theta R cos theta

so

integral d theta R cos theta pi R^2 cos^2 theta

from theta = 0 to theta = pi/2

pi R^3 d theta cos^3 theta

integral of cos^3 = sin - (1/3) sin^3

so

pi R^3 [sin theta - (1/3 sin^3 theta]

at pi/2 - at 0

pi R^3 [ 2/3]

that is for the upper hemisphere, multiply by 2 to include the bottom half

pi R^3 [4/3]

## Answer This Question

## Related Questions

- calculus *improper integrals* - A hole of a radius of 1cm is pierced in a sphere...
- Calculus - 4. At time t, t>0, the volume of a sphere is increasing at a rate ...
- calculus - 3. The radius r of a sphere is increasing at a constant rate of 0.04 ...
- Calculus - Hi there i am having some problems trying to do my calculus homework...
- college physics - Two solid spheres A and B are made from the same material. The...
- college physics - Two solid spheres A and B are made from the same material. The...
- physics - Two solid spheres A and B are made from the same material. The mass of...
- calculus - A hole of a radius of 1cm is pierced in a sphere of a 4cm radius. ...
- algebra - what is the volume of a sphere with radius r is given by the formula v...
- MAth - A sphere has a surface area of 28.26 in2. Find the diameter in inches, of...

More Related Questions