Lindsay's shape

Calculus Level 3

After studying various 3D shapes and finding formulas for their volumes, I challenged my students to invent a new shape. Lindsay created a shape ((with height 2 cm)2 \text{ cm}) that is circular at the top ((with radius 1 cm)1\text{ cm}) but square at the bottom ((with side length 2 cm).2\text{ cm}). Lindsay created this shape from a paraboloid: sliced four times parallel to the paraboloid's axis and two times perpendicular to the paraboloid's axis. Lindsay's shape is pictured below.

Find the volume of Lindsay's shape in cm3,\text{cm}^{3}, which can be written as AB+Cπ\frac{A}{B}+C\pi with A,B,CA,B,C integers, AA and BB coprime, and BB positive.

Give the value of A+B+C.A+B+C.

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