You get a bonus - 1 coin for daily activity. Now you have 1 coin

General formula for volumes of bodies of revolution

Lecture



The body of revolution in the simplest case is called such a body, which by planes perpendicular to the axis of rotation, intersect in circles with centers on this line.

  General formula for volumes of bodies of revolution

Draw a plane through the axis of the body and enter Cartesian coordinates x, y in this plane, taking the axis of the body as the axis x. The xy plane intersects the surface of the body along a line for which the x axis is the axis of symmetry. Let y = f (x) be the equation of that part of this line that is located above the x axis.
Draw a plane perpendicular to the x axis through the point (x, 0) and denote by V (x) the volume of the part of the body lying to the left of this plane; V (x) is a function of x. The difference V (x + h) - V (x) is the volume of a layer of body thickness h, enclosed between two planes that are perpendicular to the x axis and pass through points with abscissas x and x + h. Let M be the largest, and m the smallest value of the function f (x) on the segment [x, x + h]. Then the body layer under consideration contains a cylinder with radius m and height h and is contained in a cylinder with radius M and the same height h.
therefore

  General formula for volumes of bodies of revolution

When the height h tends to zero, the left and right sides of the last inequality tend to the same value πf ^ 2 (x). The middle part of this inequality as h tends to 0 tends to the derivative V` (x) of the function V (x). So

  General formula for volumes of bodies of revolution

According to the formula of analysis

  General formula for volumes of bodies of revolution

This formula gives the volume of the body part enclosed between parallel planes x = a and x = b.
created: 2014-10-05
updated: 2021-03-13
463



Rating 9 of 10. count vote: 2
Are you satisfied?:



Comments


To leave a comment
If you have any suggestion, idea, thanks or comment, feel free to write. We really value feedback and are glad to hear your opinion.
To reply

Stereometry

Terms: Stereometry