Solids of Revolution

Solids of Revolution Previously we found the areas between 2 curves. We will now take those areas and revolve them around the x or y axis, or around a 3rd linear equation. This will present us with 3 dimensional shapes that have corresponding volumes and surface areas. We will use our knowledge about integrals to evaluate the area of these figures. The same can be done to find surface area, if time allows we will investigate surface area as well.

Visual

The process of finding this figure’s volume is done by cutting the figure up into little discs of very small widths (dx). The discs will have a similar shape, they will all be cylinders. dx Find the volume of this one little disc (cylinder)

Find the volume of this one little disc (cylinder) The radius is determined by the height of your rectangle, which is the same as f(x). The height of the cylinder is dx, which in theory is infinitely small. So in theory we want to cut this ‘bullet’ up into hundreds of disks and sum all those volumes. Thus we use integration of the cylinder formula.

Lets actually integrate this thing and find the volume of the solid Lets actually integrate this thing and find the volume of the solid. Lets use limits of integration from 0 to 6. A lot of shapes that we see in the real world that were manufactured by machines use this process of finding their volumes before the factory mass produces them. Pistons for cars, bullets, gears, flanges, etc.

Find the volume of the following Might be a bad example, requires integration by parts.

Determine the volume

Add in functions revolved around y-axis, for DISC method

Discuss WASHER METHOD with X-AXIS

Since this is around the y-axis we want to integrate with respect to y Since this is around the y-axis we want to integrate with respect to y. So change the equations to be in terms of y.

Outer ring Inner ring

Because the axis of rotation is different than the x axis we must take that into account. When we identify the inner and outer radii. The inner radius would be the difference between 4 and the line y=x. So the inner radius r(x)= 4-x. Do the same thing with the outer radius. So R(x)= 4 - (x2-2x)  -x2+2x+4

It is critical to pay attention to what type of axis of revolution is being used, is it vertical or horizontal? And is it different than the x-axis or the y-axis? Revolving around the y-axis (integrate with respect to y.) Limits of integration are on y-axis Revolving around the x-axis (integrate with respect to x.) Limits of integration are on x-axis