Integration - the Australian Mathematical Sciences Institute

A guide for teachers – Years 11 and 12 • {29}has rotational symmetry about the x-axis and is called a solid of revolution.yzxWe can estimate the area under the graph y = f (x) in the plane by subdividing the x-interval and forming rectangles. Rotating these rectangles about the x-axis, we estimatethe volume of the solid of revolution by cylinders.yyyzxzxUse our usual notation: we consider the graph of y = f (x) between x = a and x = b, andwe divide the interval [a,b] into n subintervals [x 0 , x 1 ], ..., [x n−1 , x n ] of width ∆x. Aswe have seen, for the right-endpoint estimate, the rectangles have width ∆x and heightf (x j ), and hence area f (x j ) ∆x. Similarly, the cylinders have radius f (x j ) and width ∆x,and hence volume π f (x j ) 2 ∆x. We can estimate the volume of the solid of revolution as∑ nj =1 π f (x j ) 2 ∆x. Taking the limit as n → ∞, the volume of the solid of revolution is∫ baπ f (x) 2 d x.More generally, if we have a solid bounded by the two parallel planes x = a and x = b, andat each value of x in [a,b] the cross-sectional area of the solid is A(x), then the volume ofthe solid is∫ baA(x) d x.