A ball of radius 14 has a round hole of radius 7 drilled through its center. Find the volume of the resulting solid.

Answer 1

volume sphere radius of 14 = 11,494 cubic units

volume cylinder radius 7 and height 2*√147=3732.80 cubic units

still need volume of caps on top and bottom of cylinder

corrected

for top cap, integral from x=0 to x=7 of 2πx[√(196-x²)-√147]dx

using shell method

volume top cap = 2π(-((196-x²)^(3/2))/3-(x²/2)√147) from x=0 to x=7

top cap volume=148.5 cubic units

top plus bottom cap plus cylinder volumes=4029.80 cubic units

sphere minus removed cylinder and caps volume= 7464.20 cubic units

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Answer 2

See the video.

Answer 3

Draw a semicircle with radius 14, centered at the origin using y=sqrt(14² - x²), then the equation y=7. Determine the intersection of those two lines, and set the x values of ±sqrt(147) to constants a and b. Calculate the volume of the revolved solid by taking the integral from a to b of π*sqrt(196-x²)²dx. Note, this can be simplified to the 2π times the integral from 0 to sqrt(147) of (196 - x²)dx, since the semicircle is symmetric about the y axis. Calculate this volume, then subtract the volume of a cylinder with radius 7 and height 2sqrt(147) (the distance from a to b). This represents the bore hole. You should arrive at 196*π*sqrt(147) or approximately 7465.6 cubic units.