show that
a volume of sphere with radius r
V=4/3πr^3
what is the radius?
To show the formula for the volume of a sphere, we can start with the equation for the volume of a cylinder and then use calculus to integrate over the surface of the sphere.
Let's begin by considering a cylinder with radius r and height 2r, which contains the sphere. The cylinder's volume can be calculated using the formula for the volume of a cylinder:
V_cylinder = π(r^2)(2r) = 2πr^3
Now, we want to find the volume of the sphere within this cylinder. To do this, we need to integrate the height of the cylinder with respect to the radius.
We can express the radius of the sphere in terms of the height of the cylinder using the Pythagorean theorem:
r_sphere = √(r^2 - x^2)
where x is the distance from the center of the sphere to the plane parallel to the base of the cylinder.
Now, let's consider the differential volume element within the cylinder, which is a thin disk with radius x and thickness dx. The volume of this disk can be calculated as:
dV_disk = π(x^2) dx
To find the volume of the sphere, we need to integrate this expression over the entire height of the cylinder. Since the height of the cylinder is 2r, our integral limits will be -r to r:
V_sphere = ∫[from -r to r] dV_disk
= ∫[from -r to r] π(x^2) dx
Since the sphere is symmetric about the origin, the integral from -r to r will include all the positive and negative contributions, effectively doubling the volume. Therefore, we only need to compute half of the integral:
V_sphere = 2 * ∫[from 0 to r] π(x^2) dx
Using simple integration, we can evaluate this integral:
V_sphere = 2 * π * [(1/3)x^3] [from 0 to r]
= 2 * π * [(1/3)r^3 - (1/3)0^3]
= 2 * π * (1/3)r^3
Simplifying further, we find:
V_sphere = (4/3)πr^3
Hence, we have shown that the volume of a sphere with radius r is given by the formula V = (4/3)πr^3.
To prove that the volume of a sphere with radius r is equal to V=4/3πr^3, we can follow these steps:
Step 1: Start with the formula for the volume of a sphere.
The volume of a sphere is given by the formula:
V = (4/3)πr^3
Step 2: Substitute the value of r into the formula.
For a sphere with radius r, substitute r into the formula:
V = (4/3)πr^3
Step 3: Simplify the expression.
Perform the exponentiation operation to simplify the equation:
V = (4/3)πr * r * r
V = (4/3)πr^2 * r
Step 4: Use the property of exponents.
Apply the property that says (x^n)(x^m) = x^(n+m):
V = (4/3)π * r^(2+1)
Step 5: Simplify further.
Add the exponents:
V = (4/3)π * r^3
Step 6: Write in the final form.
The final expression is:
V = 4/3πr^3
Therefore, we have shown that the volume of a sphere with radius r is V = 4/3πr^3.