Al G.
asked • 07/15/21Calculus double integral
A cylindral drill with radius 2 cm is used to bore a hole through the center of a sphere with radius 7 cm. Find the volume of the ring-shaped solid that remains.
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1 Expert Answer
in line with Andrew's approach (but you don't have to explicitly get the volume of the caps): centering the sphere at the origin and the cylinder on the z-axis, note that material left remains where |z| <= √(72 - 22) = √45, and we are just removing the cylinder of volume 2 * √45 * 22π = 8√45 π = 24√5 π, and the cross-section at z has radius √(72 - z2), so area π(49 - z2), so we have I[-√45,√45]π(49 - z2) dz
I leave it to you to calculate the integral and then do the subtraction; make use of the fact that √45 = 3√5
I believe our final answer will be 180√5 π.
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Someone here may give a more efficient answer, but I would approach this by starting with the volume of the sphere and then split the drilled out hole into three sections: the right cylinder on the interior and the dome "caps" on either end. Using integration, you should be able to find the volume of each of those domes. Then you subtract the cylinder and both domes from the total spherical volume. As a side note, please be conscientious of the academic honesty policy on WyzAnt. I know it's popular to try to get free answers to graded assignments here. While we are here to help guide you in the right direction, it's unethical to request outright answers to graded assignments.07/16/21