This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 6.2: Volumes (Dated: October 1, 2011) A (right) cylinder is a solid bounded by two plane con gruent regions, say B 1 and B 2 , lying in parallel planes. It is customary to call one of these planes, say B 1 , the base of the cylinder. The height of the cylinder is the length of any of the line segments that are perpendicular to the base and join B 1 to B 2 . Let A be the area of B 1 . Then the volume of the cylinder is given by V = Ah . The volume of a solid S that isnt a cylinder can be ap proximated by a Riemann sum whose summands are vol umes of constituent cylinders. Lets clarify this. Suppose the xcoordinates of the points within S extend from x = a to x = b . Let P x be a plane perpendicular to the xaxis and passing through the point x , where a x b . The plane region obtained by intersecting S with P x is called the crosssection of S at x and has an area A ( x ). (In general, the crosssectional area A ( x ) varies as x increases from a to b .) Now lets slice the solid S into n slabs of equal width x = b a n by using the planes P a = P x , P x 1 , P x 2 , , P x n 1 , and P b = P x n , where x = a , x 1 = a + x , , x n 1 = a + ( n 1) x , x n = b . This way the i th slab S i is the part of S that lies between P x i 1 and P x i . Choosing a sample point x * i in [ x i 1 , x i ], one can approximate S...
View
Full
Document
This note was uploaded on 11/11/2011 for the course MATH 152.01 taught by Professor Geline during the Fall '09 term at Ohio State.
 Fall '09
 GELINE
 Calculus, Geometry

Click to edit the document details