A diagram would be very helpful, but I don't know how to graph in graphing calculators or computers so I cannot see the actual, precise figure. But I can "see" that the

dA = (difference in y's)*dx

As for the areas that are "negatives", if there are, I cannot see those without the figure or a sketch. Although I can do the graphs roughly by sketching them on paper here, I will not do that because the idea is to show you only the general way to do your question here.

dA is vertical. height is (y2 -y1). width is dx.

dA = [(8.5sin(0.32x -0.12) +0.56) -(4.7sin(0.42x +0.5) -4.3)]*dx

dA = [8.5sin(0.32x -0.12) -4.7sin(0.42x +0.5) +4.86)]*dx

A = INT.(-10 to 10)[8.5sin(0.32x -0.12) -4.7sin(0.42x +0.5) +4.86)]*dx

A = [(8.5/0.32){-cos(0.32x -0.12)} -(4.7/0.42){-cos(0.42x +0.5)} +(4.86)x]|(-10 to 10)

A = [-26.5625cos(0.32(10) -0.12) +11.1905cos(0.42(10) +0.5) +4.86(10)]

-[-26.5625cos(0.32(-10) -0.12) +11.1905cos(0.42(-10) +0.5) +4.86(-10)]

A = 74.9735 -(-31.9498) = 106.9233 sq.units -------answer.