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Unformatted text preview: Calculus II- Stewart Dr. Berg Spring 2010 Page 1 6.2 6.2 Volume Volume by Cross Section A right cylinder has congruent cross sections and side(s) at right angles to the base. A prism, for example, is a right triangular cylinder. If the area of a cross section is A and the height is h , then the volume is V = Ah . What if the cross-sectional area changes as the height above the base increases? Consider a cone of radius r and height h . h If we divide the height into n subintervals of height x , we can approximate the volume of a slice of the cone at height x k * in the k th interval with the volume of a right circular cylinder of radius f ( x k * ) = r h h x k * ( ) and height x = h n . The volume of the cylinder is V x k * ( ) = f x k * ( ) [ ] 2 x . Adding these volumes together, we can approximate the volume of the cone by f x k * ( ) [ ] 2 x k = 1 n = r h h x k * ( ) 2 h n k = 1 n which is a Riemann sum. Hence the actual volume would be V = lim n f x k * ( ) [ ] 2 x k = 1 n = lim n r h h x k * ( ) 2 h n k = 1 n = r h h x ( ) 2 dx h = r 2 h 2 h 2 2 hx + x 2 ( ) h dx = r 2 h...
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This note was uploaded on 02/28/2010 for the course M 56495 taught by Professor Berg during the Spring '10 term at University of Texas at Austin.
- Spring '10