In order to find the points of intersection of the line with the parabola, substitute x=7-y for x into the equation of the parabola. You will find that the points of intersection are located at (7,0) and (0,7).
Sketch a graph then realize that the height of a typical shell can be represented by 16 - (y-3)2 - (7-y) (right x - left x)
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The integral has been set up there, an answer supplied. See if you can complete the work in between.