While working out, Warren wants to know the volume of the ten weights he is lifting. Each weight is a circular disk roughly $2$ inches thick. Also, there is a circular hole (diameter $1$ inch) at the center of each weight for the barbell. If the weights were stacked from largest to smallest, the radius of each disk is $\frac { 1 } { 2 }$-inch smaller than the next larger size. (Three of the weights are shown at right.) The largest weight has a diameter of $20$ inches. The next weight has a diameter $19$ inches, etc.

1. What is the radius of the largest disk?

   Hint (a):

   The given information involves diameters, not radii.

2. What is the radius of the hole in the center?

   Hint (b):

   Refer to the hint in part (a).

3. What is the formula for the volume of the largest disk, in words?

   Hint (c):

   Each disk is a cylinder with a hole in the middle. How would you compute the volume of such a figure?

4. Write an expression for the volume of each disk if $i$ represents the number of times the radius is reduced?

   Answer (d):

   $V=\pi \left ( 10-\frac{i}{2} \right )^{2}(2)-\pi \left ( \frac{1}{2} \right )^{2}(2)$

5. Use summation notation to write an expression that will calculate the volume of the weights Warren is lifting.

   Hint (e):

   Recall that the index will account for each and every weight. How many weights are there?

6. Calculate the total volume.

   Hint (f):

   Evaluate your sigma sum.