volume of solid of revolution
Let us consider a solid of revolution, which is generated when a planar domain rotates about a line of the same plane. We chose this line for the -axis, and for simplicity we assume that the boundaries of are the mentioned axis, two ordinates , , and a continuous curve .
Between the bounds anb we fit a sequence of points and draw through these the ordinates which divide the domain in parts. Moreover we form for every part the (maximal) inscribed and the (minimal) circumscribed rectangle. In the revolution of , each rectangle generates a circular cylinder. The considered solid of revolution is part of the volume of the union of the cyliders generated by the circumscribed rectangles and at the same time contains the volume of the union of the cylinders generated by the inscribed rectangles.
Now it is apparent that
and this is true for any of the interval . The theory of the Riemann integral guarantees that there exists only one real number having this property and that it is also the definition of the integral Therefore the volume of the given solid of revolution can be obtained from
- 1 E. Lindelöf: Johdatus korkeampaan analyysiin. Neljäs painos. Werner Söderström Osakeyhtiö, Porvoo ja Helsinki (1956).
|Title||volume of solid of revolution|
|Date of creation||2013-03-22 17:20:12|
|Last modified on||2013-03-22 17:20:12|
|Last modified by||pahio (2872)|