To show that the volume of a sphere with radius [tex]\( r \)[/tex] is [tex]\( V = \frac{4}{3} \pi r^3 \)[/tex], we will use the method of integrating small volume elements.
### Step-by-Step Solution:
1. Surface Area of Sphere:
- The surface area [tex]\( A \)[/tex] of a sphere with radius [tex]\( r \)[/tex] is given by:
[tex]\[
A = 4 \pi r^2
\][/tex]
2. Small Volume Element:
- Imagine dividing the sphere into an infinite number of thin spherical shells, each with a very small thickness [tex]\( \Delta x \)[/tex].
- The radius of each small shell is [tex]\( x \)[/tex].
3. Volume of a Thin Spherical Shell:
- The volume [tex]\( \Delta V \)[/tex] of a thin spherical shell of radius [tex]\( x \)[/tex] and thickness [tex]\( \Delta x \)[/tex] is approximately the surface area at radius [tex]\( x \)[/tex] multiplied by the thickness.
- Thus, the volume [tex]\( \Delta V \)[/tex] of the shell is:
[tex]\[
\Delta V \approx 4 \pi x^2 \Delta x
\][/tex]
4. Integration to Find the Total Volume:
- To find the total volume of the sphere, we sum the volumes of all the thin shells from [tex]\( x = 0 \)[/tex] to [tex]\( x = r \)[/tex].
- This summation is represented by the integral:
[tex]\[
V = \int_{0}^{r} 4 \pi x^2 \, dx
\][/tex]
5. Evaluating the Integral:
- We need to evaluate the integral:
[tex]\[
\int_{0}^{r} 4 \pi x^2 \, dx
\][/tex]
- First, factor out the constants:
[tex]\[
V = 4 \pi \int_{0}^{r} x^2 \, dx
\][/tex]
6. Integral of [tex]\( x^2 \)[/tex]:
- Use the power rule for integration:
[tex]\[
\int x^2 \, dx = \frac{x^3}{3}
\][/tex]
- Applying this to our integral, we get:
[tex]\[
V = 4 \pi \left[ \frac{x^3}{3} \right]_{0}^{r}
\][/tex]
- Evaluate the definite integral:
[tex]\[
V = 4 \pi \left( \frac{r^3}{3} - \frac{0^3}{3} \right)
\][/tex]
[tex]\[
V = 4 \pi \left( \frac{r^3}{3} \right)
\][/tex]
[tex]\[
V = \frac{4}{3} \pi r^3
\][/tex]
### Conclusion:
Thus, the volume of a sphere with radius [tex]\( r \)[/tex] is indeed [tex]\( V = \frac{4}{3} \pi r^3 \)[/tex]. This derivation aligns perfectly with the formula for the volume of a sphere.
