In the graph below, each unit on the horizontal axis is $2\mathrm{.}1010-3$ m$3$ and each unit on the vertical axis is $0\mathrm{.}2$ atm.

On a coordinate plane where the horizontal axis is labeled V and the vertical axis is labeled P$,$ there are two paths with arrows. Both paths begin at a point labeled C$,$ and end at a point labeled A$.$ The axis units are not given, so these values will all be given merely in units. Point A is at $(6,2).$ Point C is at $(2,9).$ Point B is at $(2,2).$ Path CA is a straight line from C to A$.$ Path CBA starts at C$,$ goes down to B$,$ then right to A$.$

$($a$)$ Determine the change in internal energy of the gas if $2042$ J of heat is added as it expands in volume along the direct path CA$.$

$($b$)$ To get the same change in internal energy, how much heat must be added to the gas as it goes along the path CBA?