I think you have the region right - its boundaries are the vertical lines and on the right and left, the horizontal line on the bottom, and the curve on the top.
The way I learned it was to divide the region into vertical rectangles. These rectangles have width and height , so the sum of is the area of the region. In the limit, it's the integral of and the limits are on the left and on the right. So the area is given by:
Hopefully that answers your question.