Volume of Revolution Discussion ii

Agree that our function V(x) exists for other smooth, continuous functions

Let’s review:

When we examined the simple case, we saw our function V(b) mapped the value b to the volume of the cylinder created when the graph of the function f(x)=3 was rotated 360° around the x axis.

Our function V(b) was created using the ‘volume of cylinder’ formula; and it agrees with the formula for ‘volume of revolution’.

When we slide the value b down to zero, the volume approaches zero. When we slide the value b 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 f, when we slide b down to zero, the volume approaches zero; when we increase b the volume increases. This is still true for any negative values of f. (Try putting in f(x)=\sin(x) to the applet below.)

Not only does the volume exist for any value b in our domain; but V is increasing in a smooth and continuous way. That’s for the next page.

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