Note where the curve intersects y=-1, y=1, and y=3. The x values corresponding to these are 5, 9, and 5.
This is the graph (red) with the enclosing horizontal lines in blue and green.
The question doesn’t state what all the limits are, so I assume it’s the area between the curve, the horizontal lines and the y-axis. However, x=5 could be the left hand limit. Therefore, I’ll calculate two areas: the rectangular area between the y-axis and x=5: 20 square units. The other area is from x=5 to x=9.
Given that areas below the x axis will be calculated as negative, the area between x=5 to x=9 needs to be split into the two parts: above the axis and below the axis. However, the curve is symmetrical and the axis of symmetry is y=1, so we can find the area above y=1 and double it to get the whole area.
Area=2∫(x-5)dy=2∫(3+2y-y²)dy for y in [1,3]. We need to subtract 5 from x because we are summing the area from x=5.
Area=2[3y+y²-⅓y³] for y in [1,3].
Area=2[(9+9-9)-(3+1-⅓)=2(9-11/3)=32/3.
We may need to add in the rectangular area: 32/3+20=92/3.