You can play with your region this way:
f[x_] = Sin[10 x] + 3;
g[x_] = x^4 - 2 x^3 + x^2 + 1;
x0 = x /. FindRoot[f[x] == g[x], {x, -1}];
x1 = x /. FindRoot[f[x] == g[x], {x, 2}];
reg = ImplicitRegion[f[x] >= y >= g[x] && x0 <= x <= x1, {x, y}];
RegionPlot[reg]
RegionMeasure[DiscretizeRegion@reg]
RegionPlot[BoundaryDiscretizeRegion@reg]
RegionBoundary[BoundaryDiscretizeRegion@reg]
RegionMeasure[RegionBoundary[BoundaryDiscretizeRegion@reg]]
