Volumes of Revolution

Day 4 Volumes in Bases

The title is deceiving

This section isn’t actually rotations – instead, there will be a shape whose base will be under a curve or between two curves. We are still using the idea of V   A(x) We just will be integrating the shapes along an axis. You might see squares, semicircles, triangles… - the possibilities are endless!

Suggestion

Plan to do 2 drawings – one in the x/y plane to demonstrate the shape of the base. Then possible, a second 3D drawing that shows the shape of the figure with its base.

Demos are always great too!!

Here’s another!!

Examples

1. The vertex of a pyramid lies at the origin. The base is perpendicular to the x axis at x = 4. Cross sections are squares whose diagonals run from

y

 

5x

2

y

 

5x

2

x

section of the solid cut by a plane perpendicular to the x axis is a square with one edge in the base of the solid. Find the volume. 

y



a

3. The cross sections of a solid cut by planes perpendicular to the x axis are circles with diameter extending from

y x

between the points of intersection of these 2 curves. Find the volume of the solid.

2 to . The solid lies