Finding the volume of a solid revolved around a function or axis to create a 3D figure
The volume produced by a 3D shape which is symmetrical around the x-axis can be approximated by slicing it up into tiny cylinders, and adding them up.
We can make this approximation exact by making the cylinder infinitely small and integrating.
A curve y=f(x) can be revolved around the x-axis to produce a 3D solid. The following example shows y=2+sinx revolved 2π about the x-axis.
The volume of the resulting solid is given by
The volume of the solid produced by revolving a curve 2π about the y-axis by finding x in terms of y and evaluating
Nice work completing Volumes of Revolution, here's a quick recap of what we covered:
Exercises checked off
Finding the volume of a solid revolved around a function or axis to create a 3D figure
The volume produced by a 3D shape which is symmetrical around the x-axis can be approximated by slicing it up into tiny cylinders, and adding them up.
We can make this approximation exact by making the cylinder infinitely small and integrating.
A curve y=f(x) can be revolved around the x-axis to produce a 3D solid. The following example shows y=2+sinx revolved 2π about the x-axis.
The volume of the resulting solid is given by
The volume of the solid produced by revolving a curve 2π about the y-axis by finding x in terms of y and evaluating
Nice work completing Volumes of Revolution, here's a quick recap of what we covered:
Exercises checked off