To find the volume of a sphere using integration, we will use spherical coordinates to simplify the problem. Here's a detailed step-by-step solution:

Step 1: Understanding the Sphere's Equation

The equation of a sphere of radius $R$ centered at the origin is: $x^2 + y^2 + z^2 = R^2$

Step 2: Setting Up the Integral in Cartesian Coordinates

In Cartesian coordinates, the volume $V$ can be expressed as: $V = \iiint_{x^2 + y^2 + z^2 \leq R^2} dx\, dy\, dz$

Step 3: Switching to Spherical Coordinates

We switch to spherical coordinates, where: $x = \rho \sin \theta \cos \phi$ $y = \rho \sin \theta \sin \phi$ $z = \rho \cos \theta$

The volume element $dV$ in spherical coordinates is given by: $dV = \rho^2 \sin \theta \, d\rho \, d\theta \, d\phi$

Step 4: Setting Up the Limits of Integration

In spherical coordinates, the limits are:

• $\rho$ ranges from 0 to $R$
• $\theta$ ranges from 0 to $\pi$
• $\phi$ ranges from 0 to $2\pi$

Step 5: Writing the Integral

The volume integral becomes: $V = \int_0^{2\pi} \int_0^{\pi} \int_0^R \rho^2 \sin \theta \, d\rho \, d\theta \, d\phi$

Step 6: Evaluating the Integral

1. Integrate with respect to $\rho$: $\int_0^R \rho^2 \, d\rho = \left[ \frac{\rho^3}{3} \right]_0^R = \frac{R^3}{3}$

2. Integrate with respect to $\theta$: $\int_0^{\pi} \sin \theta \, d\theta = \left[ -\cos \theta \right]_0^{\pi} = (-\cos(\pi)) - (-\cos(0)) = 1 + 1 = 2$

3. Integrate with respect to $\phi$: $\int_0^{2\pi} d\phi = \left[ \phi \right]_0^{2\pi} = 2\pi$

Step 7: Combining the Results

Combining these results, we get: $V = \left( \frac{R^3}{3} \right) \cdot 2 \cdot 2\pi = \frac{2R^3}{3} \cdot 2\pi = \frac{4\pi R^3}{3}$

Final Result

The volume $V$ of a sphere of radius $R$ is: $V = \frac{4\pi R^3}{3}$

Would you like a detailed explanation of any part, or have any questions about the steps? Here are some related questions you might be interested in:

1. How does the volume formula for a sphere change if the radius is doubled?
2. What are the steps for converting integrals from Cartesian to spherical coordinates?
3. Can we use cylindrical coordinates to find the volume of a sphere? How?
4. How do we derive the volume of a sphere using the disk method?
5. How do spherical coordinates relate to Cartesian coordinates?
6. Can you explain the importance of the Jacobian determinant in coordinate transformations?
7. How can we use integration to find the surface area of a sphere?
8. What are some practical applications of finding the volume of a sphere?

Tip: When solving volume integrals, choosing the appropriate coordinate system can simplify the calculations significantly, especially for symmetrical objects like spheres.