Çözüldü Applications of Integration-Hexagonal Frustum Lantern's Volume-Cavalieri's Principle-Radical Numbers

A slightly changed version of a question without a solution from Harvard University:

https://i.ibb.co/3hZXy2h/Harvard.png

What is the minimum volume, in cubic units, of the lamp shown in the question above? (No calculator is allowed.)

A) 200
B) 202
C) 204
D) 206
E) 208


https://i.ibb.co/BNf5wZN/Harvard-WA.png
https://www.wolframalpha.com/input?i=integrate 2(1+sqrt(2))z(1+z)^2dz from z=0..3

Note: Doing the integration and knowing the inequality that √2 > 1.41 are the interested students' responsibilities.
Reference: https://mathbitsnotebook.com/Geometry/3DShapes/3DCavalieri.html
Original Question: https://people.math.harvard.edu/~knill/teaching/math1a2020/final/practice1.pdf (Page 5, Problem 5)