A particular commodity has a price$-$supply equation given by $p=362(1\mathrm{.}037{)}^x,$ where $x$ is the numbers of items of the commodity demanded when the price is $p$ dollars per item.

$($a$)$ Find producers' surplus if the equilibrium quantity is $56$ items. $($Round your answer to the nearest cent if necessary.$)$

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$($b$)$ Find producers' surplus if the equilibrium price is $2,234$ dollars. $($Round your answer to the nearest cent if necessary.$)$