To determine
To find:
Integral for the volume of cut out
Answer
V=8∫0rr2-y2·R2-y2dy
Explanation
1) Concept:
i) Solid of Revolution:
Volume of solid revolution revolving a region around a line is given by
V=∫abA(x)dx or V=∫cdA(y)dy
In case of the disk method, the radius is found in terms of x or y and use A=π·radius2
In case of the washer method, find the inner and outer radius and compute the area of washer by subtracting the area of inner disk from outer disk, use
A=π·outer radius2-π·inner radius2
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:

Now from the front side, the circle appears to be in a y-z plane,
Therefore, equation of circle is y2+z2=R2
From the top side, the hole lies on a x-y plane,
Therefore, equation of circle is x2+y2=r2
From the front side, the diagram will appear as shown below:

Here, the x- axis is the axis of cylindrical hole of radius r. A quarter of the cross section through y, perpendicular to the y- axis, is the rectangle shown.
By using Pythagoras theorem,
x=r2-y2 and z=R2-y2
So,
14·Ay=xz
Ay=4r2-y2·R2-y2
V=∫abA(y) dy
Since the cut out varies from –r to r
V=∫-rr4r2-y2·R2-y2dy
V=4·∫-rrr2-y2·R2-y2dy
By using concept of integrals of symmetric function,
V=8∫0rr2-y2·R2-y2dy
So, the above value obtained is the volume of thecut out portion.
Conclusion:
V=8∫0rr2-y2·R2-y2dy