A little bit of both

Calculus Level 5

For which real numbers \(a\) and \(b\) with \(0\leq{a}\leq{b}\leq{1}\) is \[\int_{a}^{b}\left(\cos(7\pi{x})-\cos(9\pi{x})\right) \ dx\] maximal? Find \(A=a\times b\).

No calculators, please; they spoil all the fun.

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