Chapter 5, Problem 7ETE

Now consider the bottle shown at the right. To find the bottle’s volume in a similar manner, we turn it on its side, use a measuring device to take vertical measurements of the radii, and proceed as we did with the urn.

The table of measurements is as follows.

x (in inches)	y (in inches)
0	1.125
1	1.275
2	1.250
3	1.275
4	1.275
5	1.125
6	1.000
7	0.875
8	0.750
9	0.500
10	0.500

To determine the volume of the bottle’s base, extend the x-axis 0.015 unit to the left of 0. The total volume of the plastic that forms the bottle can then be found by first integrating $y_3=\pi {\left(y_2\right)}^2$ over $\left[-0.015,0\right]$ to find the volume of the base and then integrating $y_4=\pi {\left(y_2-y_1\right)}^2$ over $\left[0,10\right]$ to find the volume of the side. What is the total volume of plastic used to form the bottle?