How much present did you get for Christmas?

Jelly, 19 18 bytes

Zµ*3×1420÷339Ḣo@PS

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Unfortunately, Jelly does not have a π constant yet, and the vectorizer doesn't handle floats properly.

To overcome these issues, instead of multiplying by 4π/3, we multiply by 1420 and divide by 339. Since 1420 ÷ 339 = 4.18879056… and 4π/3 = 4.18879020…, this is sufficiently precise to comply with the rules.

The newest version of Jelly could accomplish this task in 14 bytes, with better precision.

Zµ*3×240°Ḣo@PS

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How it works

Zµ*3×1420÷339Ḣo@PS  Left argument: A, e.g., [[1, 2, 3], [4, 0, 0]]

Z                   Zip A; turn A into [[1, 4], [2, 0], [3, 0]].
 µ                  Begin a new, monadic chain with zip(A) as left argument.
  *3                Cube all involved numbers.
    ×1420           Multiply all involved numbers by 1420.
         ÷339       Divide all involved numbers by 339.
                    This calculates [[4.19, 268.08], [33.51, 0], [113.10, 0]]
             Ḣ      Head; retrieve the first array.
                    This yields [4.19, 268.08].
                P   Take the product across the columns of zip(A).
                    This yields [6, 0].
              o@    Apply logical OR with swapped argument order to the results.
                    This replaces zeroes in the product with the corresponding
                    results from the left, yielding [6, 268.08].
                 S  Compute the sum of the resulting numbers.

The non-competing version uses ×240° instead of ×1420÷339, which multiplies by 240 and converts the products to radians.

Pyth, 19 18 bytes

sm|*Fd*.tC\ð7^hd3Q

1 byte thanks to Dennis

Demonstration

Input format is list of lists:

[[1,4,3],[2,2,2],[3,0,0],[4,4,4]]

It simply multiplies the dimensions together to calculate the cube volume. If that comes out to zero, it calculates the sphere volume.

The sphere constant, 4/3*pi is calculated as 240 degrees in radians. .t ... 7 converts an input in degrees to radians, and C\ð calculates the code point of ð, which is 240.

Haskell, 40 bytes

q[x]=4/3*pi*x^^3
q x=product x
sum.map q

Usage example: sum.map q $ [[1,4,3],[2,2,2],[3],[4,4,4]] -> 197.09733552923254.

How it works: For each element of the input list: if it has a single element x calculate the volume of the sphere, else take the product. Sum it up.