Agree that our function V(x) exists for other smooth, continuous functions
When we examined the simple case, we saw our function mapped the value to the volume of the cylinder created when the graph of the function was rotated 360° around the axis.
Our function was created using the ‘volume of cylinder’ formula; and it agrees with the formula for ‘volume of revolution’.
When we slide the value down to zero, the volume approaches zero. When we slide the value to the right, the volume increases smoothly.
The single purpose on this page is to agree that if we have a smooth and continuous function , when we slide down to zero, the volume approaches zero; when we increase the volume increases. This is still true for any negative values of . (Try putting in to the applet below.)
Not only does the volume exist for any value in our domain; but V is increasing in a smooth and continuous way. That’s for the next page.
Next Page: V increases in a smooth and continuous way
Volume of Revolution menu