Find the volume of the solid obtained by rotating the region bounded by y= x^2, y=0, and x=3 about the x-axis.
To find the volume of the solid obtained by rotating the region bounded by the curves y = x^2, y = 0, and x = 3 about the x-axis, we can use the method of cylindrical shells.
First, let's draw a graph of the given region:
```
| /
| /
| /
| /
| /
______|/_____
```
The region is bounded by the curve y = x^2, the x-axis, and the vertical line x = 3.
To find the volume, we divide the region into infinitesimally thin vertical slices or "shells" and find the volume of each shell. The volume of each shell can be approximated as the product of its height (the difference in x-values) and circumference (2π times the x-value).
Let's consider a shell located at a distance x from the y-axis. The height of this shell will be 'dy' and the circumference will be 2πx.
The formula for the volume of each cylindrical shell is:
V = 2πx * dy
To express the height of the shell 'dy' in terms of x, we rearrange the equation of the parabola y = x^2 as:
y = x^2
dy = 2x * dx
Now, we can substitute dy and the limits of integration into the volume formula:
V = ∫[0,3] 2πx * dy
V = ∫[0,3] 2πx * (2x * dx)
V = 2π ∫[0,3] x^2 * dx
Evaluating this definite integral from x = 0 to x = 3 will give us the volume of the solid obtained by rotating the region bounded by y = x^2, y = 0, and x = 3 about the x-axis.