# calculus

posted by .

Use integrals to prove that the volume of a sphere of radius
R is equal to (4/3)(pi)R^3

• calculus -

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 -

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]

## Similar Questions

1. ### Calculus

Hi there i am having some problems trying to do my calculus homework and i really need help on how to show the step to proof the volume of a sphere which is V= 4/3pirsquare. But I have to use triple integral to proof the volume of …
2. ### physics

Two solid spheres A and B are made from the same material. The mass of sphere B is eight times that of sphere A. If the radius of sphere A is 7.50 cm, what is the radius of sphere B?
3. ### college physics

Two solid spheres A and B are made from the same material. The mass of sphere B is eight times that of sphere A. If the radius of sphere A is 7.50 cm, what is the radius of sphere B?
4. ### college physics

Two solid spheres A and B are made from the same material. The mass of sphere B is eight times that of sphere A. If the radius of sphere A is 7.50 cm, what is the radius of sphere B?
5. ### algebra

what is the volume of a sphere with radius r is given by the formula v=4/3 r^3 Find the volume of sphere with radius 4 meters. use 3.14 for the value of x and round your answer to the nearest tenth.
6. ### calculus

3. The radius r of a sphere is increasing at a constant rate of 0.04 centimeters per second. (Note: The volume of a sphere with radius r is v=4/3pir^3 ). a. At the time when the radius of the sphere is 10 cm, what is the rate of increase …
7. ### Calculus

4. At time t, t>0, the volume of a sphere is increasing at a rate proportional to the reciprocal of its radius. At t = 0, the radius of the sphere is 1 and at t = 15 the radius is 2. a. Find the radius of the sphere as a function …
8. ### calculus

A hole of a radius of 1cm is pierced in a sphere of a 4cm radius. Calculate the volume of the remaining sphere. I know that the volume of a sphere with an exterior radius is of: V = pie(R^2 - r^2) dH where R^2 is the exterior surface …
9. ### calculus *improper integrals*

A hole of a radius of 1cm is pierced in a sphere of a 4cm radius. Calculate the volume of the remaining sphere. I know that the volume of a sphere with an exterior radius is of: V = pie(R^2 - r^2) dH where R^2 is the exterior surface …
10. ### Calculus

Imagine slicing through a sphere with a plane (sheet of paper). the smaller piece produced is called a radius of the sphere. Its volume is V=(pi)h^2(3r-h)/3, where r is the radius of the sphere and h is the thickness of the cap. find …

More Similar Questions