Consider a right circular cylinder with radius $ r$ of the base, hight $ h$. Find the volume of the solid by revolving the cylinder about a diameter of the base.
To find the volume of the solid formed by revolving a right circular cylinder about a diameter of its base, we need to use the method of cylindrical shells or the method of disks/washers. Here, we will use the method of cylindrical shells.

1. **Set up the problem:**
   - The cylinder has a radius \( r \) and height \( h \).
   - We revolve the cylinder about a diameter of the base, which we can take as the \( x \)-axis.

2. **Consider a thin vertical slice of the cylinder:**
   - A vertical slice at a distance \( y \) from the center of the base has a thickness \( dy \).
   - The height of this slice is \( h \).
   - The radius of rotation for this slice is \( y \).

3. **Volume of a cylindrical shell:**
   - The volume of a thin cylindrical shell is given by the formula:
     \[
     dV = 2\pi \cdot (\text{radius of shell}) \cdot (\text{height of shell}) \cdot (\text{thickness of shell})
     \]
   - Here, the radius of the shell is \( y \), the height of the shell is \( h \), and the thickness of the shell is \( dy \).
   - Therefore, the volume of the shell is:
     \[
     dV = 2\pi y h \, dy
     \]

4. **Integrate to find the total volume:**
   - The shells extend from \( y = -r \) to \( y = r \).
   - Thus, the total volume \( V \) is:
     \[
     V = \int_{-r}^{r} 2\pi y h \, dy
     \]

5. **Evaluate the integral:**
   - Notice that the integrand \( 2\pi y h \) is an odd function (since it is proportional to \( y \)), and we are integrating it over a symmetric interval \([-r, r]\).
   - The integral of an odd function over a symmetric interval around zero is zero:
     \[
     \int_{-r}^{r} y \, dy = 0
     \]
   - Therefore:
     \[
     V = 2\pi h \int_{-r}^{r} y \, dy = 2\pi h \cdot 0 = 0
     \]

6. **Reconsider the problem:**
   - The above result suggests that the volume calculation using cylindrical shells is not appropriate here. Instead, we should use the method of disks/washers.

7. **Using the method of disks/washers:**
   - Consider a horizontal slice of the cylinder at height \( z \) from the base.
   - The radius of this slice is \( r \).
   - The thickness of the slice is \( dz \).

8. **Volume of a disk:**
   - The volume of a thin disk is given by the formula:
     \[
     dV = \pi (\text{radius})^2 (\text{thickness})
     \]
   - Here, the radius of the disk is \( r \), and the thickness is \( dz \).
   - Therefore, the volume of the disk is:
     \[
     dV = \pi r^2 \, dz
     \]

9. **Integrate to find the total volume:**
   - The disks extend from \( z = 0 \) to \( z = h \).
   - Thus, the total volume \( V \) is:
     \[
     V = \int_{0}^{h} \pi r^2 \, dz
     \]

10. **Evaluate the integral:**
    \[
    V = \pi r^2 \int_{0}^{h} dz = \pi r^2 [z]_{0}^{h} = \pi r^2 (h - 0) = \pi r^2 h
    \]

The final answer is \(\boxed{\pi r^2 h}\).