!!

Volume, Area, Length

AREA BETWEEN AND

Example Find the area of the region enclosed by and

Solution These curves intersect at (x,y) satisfying:

points of intersection and (4, 2)

VOLUMES BY SLICING

For a cylinder

cross-sectional area at is

Approximate volume is

SOLIDS OF REVOLUTION

From: http://www.pro-nutrition.co.uk/fees.htm

Example

Find the volume of the solid generated when the region between the graphs of the equations and over the interval [0, 2]is revolved about the x-axis

Solution

LENGTH OF A PLANE CURVE

AREA OF A SURFACE OF REVOLUTION

From: http://byteshuffler.com/rospo/blog/2006/11/apple-peeler-slicer-corer-gizmo.html

From: http://byteshuffler.com/rospo/blog/2006/11/apple-peeler-slicer-corer-gizmo.html

the lateral area of the kth frustum is

Rotating about the y-axis produces:

ExampleFind the area of the surface that is generated by revolving the portion of the curve

between and about the x-axis.