Find the volme of the solid generated by revolving the region bounded by the triangle with vertices (1,1) (1,3) and (2,3) about the x axis using shells and washers.
To find the volume of the solid generated by revolving the region bounded by the triangle about the x-axis, we can use both the method of shells and the method of washers.
First, let's consider the method of shells:
1. Draw a diagram of the triangle and the axis of revolution (in this case, the x-axis).
2. Slice the triangle into thin vertical strips or shells, perpendicular to the x-axis.
3. Each shell has a thickness dx and a height equal to the difference between the upper and lower y-values at that x-coordinate.
4. The volume of each shell is given by the formula V_shell = 2πx * (height) * dx.
5. Integrate the volume of each shell over the interval of x-values that covers the triangle to find the total volume.
Now, let's move on to the method of washers:
1. Draw a diagram of the triangle and the axis of revolution (x-axis).
2. Slice the region into thin horizontal washers, parallel to the x-axis.
3. Each washer has a thickness dx and its inner and outer radii are determined by the distance between the x-axis and the upper and lower boundary curves, respectively.
4. The volume of each washer is given by the formula V_washer = π * (outer radius squared - inner radius squared) * dx.
5. Integrate the volume of each washer over the interval of x-values that covers the triangle to find the total volume.
Both methods will give you the same result, so you can choose whichever method you find more comfortable or convenient to use.