The volume of solid S, where the base of S is a circular disk with radius r and parallel cross sections perpendicular to the base are squares.
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
The equation of the circle is x2+y2=r2
The upper semicircle has equation yu= r2-x2 and
The lower semicircle has equation: yl=-r2-x2
From figb, it can be seen that side length of the square is the difference between yu and yl.
Therefore, the area of the cross section is
After substituting the values of yu and yl.
From the figure, it can be seen that limit is varying from –r to r.
Therefore, from definition of Volume
Substitute the value of Ax,
Applying the Fundamental Theorem of Calculus,
The volume of solid S where the base of S is a circular disk with radius r and parallel cross sections perpendicular to the base are squares is 16r33