The volume of the circular cone with the height h and the base radius r.
i. If x is the radius of the typical shell, then the circumference =2πx and the height is y.
ii. By the shell method, the volume of the solid by rotating the region under the curve y=f(x) about y- axis from a to b is
V= ∫ab2πx f(x)dx
The right circular cone with the height h and the base radius r
The right circular cone with the height h and the base radius r is as shown in the figure. It is obtained by rotating a triangle joining (0,0), (0, h) and (r,0) about y-axis
From the figure, write the equation of line passing through the two points (r, 0) and (0, h).
The slope of the line is =h-00-r The y-intercept is h
Therefore, the equation of the line is y=-hrx+h, and it rotates about y-axis.
Using the shell method, find the typical approximating shell with the radius x.
Therefore, the circumference is 2πx and the height is y=-hrx+h
So, the total volume is
Therefore, the volume of the circular cone of the height h and the radius r is V=πr2h3unit3
The volume of the cone of the height h and the radius r is