#### To determine

**To find:**

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.

#### Answer

16r33

#### Explanation

**1) Concept:**

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

V=limn→∞∑i=1nAxi*∆x=∫abAxdx

**2) Calculations:**

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

Ax= yu-yl2

After substituting the values of yu and yl.

Ax=r2-x2--r2-x2 2=2r2-x22=4(r2-x2)

From the figure, it can be seen that limit is varying from –r to r.

Therefore, from definition of Volume

V=∫-rrAxdx

Substitute the value of Ax,

V=∫-rr4r2-x2dx

After integrating,

V=4r2x-x33r-r

Applying the Fundamental Theorem of Calculus,

V=4r2·r-r33-4(r2(-r)--r33)

After simplification,

V=4r3-r33-4-r3+r33=42r3-r33=16r33

**Conclusion:**

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