Final answer:
To find the volume of the solid created by rotating a triangle around the y-axis, use the Cylindrical Shells method to set up and evaluate an integral for each of the two linear segments forming the triangle's sides.
Step-by-step explanation:
To calculate the volume of the solid generated by rotating a triangle around the y-axis, we need to choose an appropriate rotational volume method. For the triangle with vertices (1,2), (2,0), and (3,2), it is convenient to use the Cylindrical Shells method. First, we should express the triangle's linear equations as functions of x, which will serve as the height of each cylindrical shell. In this case, there are two functions, because the triangle has two sides that are non-vertical. One side can be expressed by the equation for a line passing through (1,2) and (2,0), and the other through (2,0) and (3,2).
The volume of the whole solid is found by calculating the volume of the individual shells and integrating these volumes from x = 1 to x = 3, the limits that span the base of the triangle along the x-axis.
To set up the integral, whether using shell height or the radius (distance from the y-axis), we need to recognize the radius of each shell which is simply 'x', since we are rotating around the y-axis. Then, applying the volume of a cylinder formula V = πr²h, we integrate the volume of each 'shell' which is 2πrh where r is the shell radius and h is the shell height (the value of the triangle's linear equations). The exact integral will differ depending on which section of the triangle we're considering (because the triangle's sides have different slopes).
The final integral will look like this for one of the sections:
V = ∫ (2πx * height of the shell) dx, from x = 1 to x = 2,
And a similar integral from x = 2 to x = 3 for the other section.
Summing up both integrals will give the total volume of the solid obtained by rotating the triangular area around the y-axis.