The volume of a frustrum of a right circular cone with height h, lower base radius R, and top radius r.
i. Use the definition of volume
ii. A frustum of a cone is the part of the cone that remains after the top of the cone is cut-off parallel to the base of the cone.
2) Definition of volume:
Let S be a solid that lies between x=a and x=b. If the cross sectional area of S in the plane Px, through x and perpendicular to the x-axis, is A(x), where A is a continuous function, then the volume of S is
Height is h, lower base radius is R, top radius is r.
Since the frustum has rotational symmetry, the above region shall produce the frustum upon rotation about x-axis. Consider a representative rectangle in it .
The slope of the line with end points (h, r) and (0, R) is r-Rh-0=r-Rh, and the y-intercept is R.
Using the slope intercept form, the equation of the line that represents the side of the frustum is y=r-Rhx+R.
The figure represents the disk (washer with no hole) that is generated from revolving the representative rectangle about x-axis.
The radius is,
So, the volume of the solid of revolution that is the volume of the frustum is,
Vfrustum=∫abAxdx =∫x=0x=h πr-Rhx+R2 dx=∫x=0x=h πr-R2hx2+2R·r-Rhx+R2 dx
By applying the Fundamental Theorem of Calculus,
πr-R2h2·h33 +2R·r-Rh·h22+R2h-πr-R2h2·033 +2R·r-Rh·022+R20
After expanding the term inside bracket,
After cancelling the common terms and simplifying the equation,
The volume of a frustum of a right circular cone with top radius r, bottom radius R and height h is