Bounds and Bounds

Calculus Level 4

Assume that the equation \(\displaystyle \int _{ 2 }^{ \sqrt [ 3 ]{ B } }{ \frac { { 12x }^{ 2 } }{ \sqrt { { x }^{ 3 }+1 } }\, dx=200 } \) holds true where \(B=Q\times R\) and \(Q\) and \(R\) are coprime integers. If \(\displaystyle \int _{ Q }^{ R }{ (-x+2)\, dx=52 } \), what is the value of \(R\)?

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