Finding Volumes by Integration Section 7.2

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].

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. 

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

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

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

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!