1. Finding Volumes by Integration
Section 7.2

2. General Volume Formula
Let S be a solid that lies between x = a and x = b, and let the cross-sectional area of S perpendicular to the x-axis at a given value of x be represented by A(x), where A(x) is a continuous function. Then the volume of S is given by
where ci is a sampling point on the subinterval [xi-1, xi].

3. Solids of Revolution
Consider a region bounded by some function f (x) and the x-axis, between x = a and x = b. By rotating the curve of f (x) 360º about the x-axis and filling the region traced we obtain a solid of revolution. 

4. Solids of Revolution Nice Properties
Symmetry about the axis of rotation
Perpendicular cross-sections are…
Circular Disks OR
Circular Washers

5. Volume Formula for Solids of Revolution -Disk Method-
Let S be a solid of revolution formed by rotating the curve f (x) between x = a and x = b about the x-axis. Then the volume of S is given by

6. Volume Formula for Solids of Revolution -Washer Method-
Let S be a solid of revolution formed by rotating the area bounded by the curves f (x) from above and g (x) from below about the x-axis (on the interval [a, b]). Then the volume of S is given by

7. Rotating About Non-Axis Lines
It’s perfectly acceptable to rotate about the lines x = a or y = b (where a and b are non-zero).
Use the disk or washer method as appropriate.
Think carefully about the radius (or radii) of the resulting solid!