A solid has at its base the region bounded by the parabola x-y^2 = -8 and the left branch of the hyperbola x^2-y^2-4=0. The vertical slices perpendicular to the x-axis are squares. Find the volume of the solid. This is the full problem.
I was able to find out the points of intersection, which were x=-3 and x=4. I am not sure what the cross-sectional area is.
the left side of the hyperbola is
You've identified the x co-ordinate of the points of intersection.
x=4 is unnecessary, as the parabola intersects the rightside part there.
To calculate the area of the bounded region, you only need the area of the part above the x-axis and multiply by 2.
If you also calculate where the parabola and hyperbola cut the x-axis to the left of the y-axis, then you can integrate under the parabola from x=-8 to -3
and integrate under the hyperbola from x=-3 to x=-2.
That will be one way to calculate the area between the curves.
I can't visualise the object itself however, as your question mentions a square.