#### To determine

**To find:**

Volume common to two circular cylinders

#### Answer

V=16r33

#### Explanation

**1) Concept:**

i) Let S be a solid that lies between x=a to 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 continuous function, then the volume of S is

V=limn→∞∑i=1nA(xi*)∆x=∫abA(x)dx

ii) Integrals of Symmetric functions:

Suppose f is continuous on [-a, a]

a) If f is even [f-x=fx], then ∫-aaf(x)dx=2∫0af(x)dx

b) If f is odd [f-x=-fx], then ∫-aaf(x)dx=0

**2) Calculation:**

If we slice the figure below, square is formed connecting four corner points. So, the cross sectional area is a square.

Looking from the shaded region as shown above, the following figure can be obtained. The side of square at y is a chord connecting diametrically opposite points so,

Since AB is the side of square, so total length of AB=2x

Equation of circle, centered at origin of radius r is x2+y2=r2

x2=r2-y2

x=r2-y2

AB=2x=2r2-y2

Now area of cross section is

A=side2=(AB)2

(AB)2=2r2-y22=4(r2-y2)

The range of y is -r, r

V=∫abA(y)dy=∫-rr4(r2-y2)dy

By using concept of integrals of symmetric function

V=2∫0r4(r2-y2)dy

V=8∫0r(r2-y2)dy

V=8∫0rr2dy-∫0ry2dy

V=8r2y0r-y330r

V=8r3-r33

V=83r33-r33

V=82r33

V=16r33

**Conclusion:**

V=16r33