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6.2 Approximating Volume with Cross-sections The volume V of a (right) cylinder is A Â· h , where A is the area of the base (and top). r base A = Ï€r 2 h Non-cylinders can be sliced into cross-sections. Each cross-sectional piece of a solid can be approximated by a cylinder. Then the volume of the body is approximated by the sum of the volumes of cylinders. 1
6.2 Volume as an Integral of Cross-sectional Areas Suppose a body is cut into N horizontal cross-sections of thickness Î” y . At each height y i , let A ( y i ) be the area of the cross-section. Each cross-section has volume approximately A ( y i )Î” y . The volume is approximated by the sum of these volumes: V â‰ˆ N X 1 A ( y i )Î” y And as N â†’ âˆž , the error in this approximation becomes arbitrarily small. The sum becomes an integral, and we find the volume: Volume of body = lim N â†’âˆž N X 1 A ( y i )Î” y = Z b a A ( y ) dy where the solid body extends from y = a to y = b . 2
6.2 Volume as an Integral: Example Calculate the volume of a pyramid of height 9 whose base is a square with side length 3.