A curve \(C\) has equation \(y = 2\mathrm{e}^{x^3} + a\), where \(a\) is a positive constant. The point \(T\) lies on \(C\) and has an \(x\)-coordinate of \(1\).
Use calculus to find the equation of the tangent to \(C\) at \(T\). Give the equation in the form \(y = \mathrm{e}(px + q) + ra\), where \(p\), \(q\) and \(r\) are exact constants to be found.[5]
It is given that the tangent to \(C\) at \(T\) passes through the origin.
Find the exact value of \(a\).[1]
Find the area of the region bounded by \(C\), the line \(x = 0\) and the tangent at \(T\). Give your answer correct to \(1\) decimal place.[2]