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Calculus II Notes 6.2

# Calculus II Notes 6.2 - Calculus II-Stewart Dr Berg Spring...

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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 Δ 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 Δ x = h n . The volume of the cylinder is V x k * ( ) = π f x k * ( ) [ ] 2 Δ x . Adding these volumes together, we can approximate the volume of the cone by π f x k * ( ) [ ] 2 Δ x k = 1 n = π 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 →∞ π f x k * ( ) [ ] 2 Δ x k = 1 n = lim n →∞ π r h h x k * ( ) 2 h n k = 1 n = π r h h x ( ) 2 dx 0 h = π r 2 h 2 h 2 2 hx + x 2 ( ) 0 h dx = π r 2 h 2 h 2 x hx 2 + x 3 3 0 h = π r 2 h 2 h 2 h ( ) h h ( ) 2 + h ( ) 3 3

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Calculus II Notes 6.2 - Calculus II-Stewart Dr Berg Spring...

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