4 UNIT MATHEMATICS 5.5 SOLIDS OF REVOLUTION (Part 3) Cylindrical Shell Method Let be a continuous functions such that R The region is the area bounded by the function and the X-axis in the [a, Revolving b]. R interval of the region about the Y-axis generates a solid R x called a solid of revolution. Consider a thin strip of the region at position x and of width and height . When this strip is rotated around the Y-axis it and x, x sweeps out a cylindrical shell of radius thickness height The volume . of this thin cylindrical shell is x By letting summing the contribution of each shell for in the interval , the and V volume of the solid of revolution is given by the definite integral cylindrical shell formula Example
4 UNIT MATHEMATICS 5.5 SOLIDS OF REVOLUTION (Part 3) Cylindrical Shell Method Let be a continuous functions such that The region R is the area bounded by the function and the X-axis in the interval [a, b]. Revolving of the region R about the Y-axis generates a solid called a solid of revolution. Consider a thin strip of the region R at position x and of width x and height . When this strip is rotated around the Y-axis it sweeps out a cylindrical shell of radius x, thickness x and height . The volume of this thin cylindrical shell is By letting and summing the contribution of each shell for x in the interval , the volume V of the solid of revolution is given by the definite integral cylindrical shell formula Example