Looks Like Gabriel's Horn To Me

Calculus Level 5

An infinitely long cone can be generated by revolving the following curve about the xx-axis.

y=ex,x<0y = e^{x}, \quad x < 0

If the ratio of volume to lateral surface area of the cone is β\beta , then what is 104×β \lfloor 10^4 \times \beta \rfloor ?

Notation: \lfloor \cdot \rfloor denotes the floor function.

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