Find the volume of a solid if the base of the solid is the circle given by the equation \[{x^2} + {y^2} = 1,\] and every perpendicular cross section is a square.
Solution.
An arbitrary cross section at a point \(x\) has the side \(a\) equal to
Find the volume of the frustum of a cone if its bases are ellipses with the semi-axes \(A, B,\) and \(a, b\), and the altitute is equal to \(H.\)
Solution.
The volume of the frustum of the cone is given by the integral
\[V = \int\limits_0^H {A\left( x \right)dx} ,\]
where \({A\left( x \right)}\) is the cross-sectional area at a point \(x.\)
The lengths of the major and minor axes linearly change from \(a, b\) to \(A, B,\) and at the point \(x\) they are determined by the following expressions:
A wedge is cut out of a circular cylinder with radius \(R\) and height \(H\) by the plane passing through a diameter of the base (Figure \(10\)). Find the volume of the cylindrical wedge.
Solution.
A cross section of the wedge perpendicular to the \(x-\)axis is a right triangle \(ABC.\) The leg of the triangle \(AB\) is given by
\[AB = y = \sqrt {{R^2} - {x^2}} ,\]
and the other leg \(BC\) is expressed in the form
\[BC = AB \cdot \tan \alpha = AB \cdot \frac{H}{R} = \frac{H}{R}\sqrt {{R^2} - {x^2}}.\]
Hence, the area of the cross section is written as
of the total volume of the cylinder. The result does not depend on \(R\) and \(H!\)
Example 10.
The axes of two circular cylinders with the same radius \(R\) intersect at right angles. Find the volume of the solid common to both these cylinders.
Solution.
The figure below shows \(\frac{1}{8}\) of the solid of intersection.
Consider a cross section \(ABCD\) perpendicular to the \(x-\)axis at an arbitrary point \(x\). Due to symmetry, the cross section is a square with sides of length
\[BC = AD = y = \sqrt {{R^2} - {x^2}} ,\]
\[AB = CD = z = \sqrt {{R^2} - {x^2}}.\]
The cross-sectional area is expressed in terms of \(x\) as follows: