first, graph the two equations (in one cartesian plane)
then get the points of intersection:
x+7=y^2 *this is the second equation
substitute this to the first:
y=-7 and y=3
substitute these back to obtain corresponding values of x:
therefore, points of int are (42,-7) and (2,3)
looking at the graph, i suggest you do vertical strips (that is, dx),, divide the whole area into region 1 (left side) and region 2 (right side),, after you do this, get the boundaries.
for x: the boundaries are the graph of the parabola (that is x=y^2-7) and the x-coord of the first point of int (x=2)
for y: since parabola is symmetric with respect to x-axis, y is from -3 to 3.
the area of region 1 is:
integral[from -3 to 3](integral[from y^2-7 to 2] dx)dy)
*note: this is double integral since A=dxdy
for region 2, i suggest you do horizontal strips (that is, dy) ,,then get the boundaries:
for x: from x-coord of first point of int (x=2) to x-ccord of 2nd point of int (x=42)
for y: from the parabola (y=sqrt(x+7)) to the line (y=(14-x)/4)
the area of region 2 is:
integral[from 2 to 42](integral[from sqrt(x+7) to (14-x)/4] dy)dx)
i'll leave the integration calculation to you,, add the areas and you'll finally get the whole area.
so there,, sorry for long explanation,,
i hope i was able to help,, =)