Sketch the region in the first quadrant enclosed by y=7/x, y=2x, and y=x/2. Decide whether to integrate with respect to x or y. Then find the area of the region.

Answer 1

It doesn't really matter which variable is used, since the boundary changes in either direction.

Cleverly enough, the hyperbola intersects the two lines at (√(7/2),√14) and (√14,√(7/2).

So,

a = ∫[0,√(7/2)] (2x)-(x/2) dx
+ ∫[√(7/2),√14] (7/x)-(x/2) dx

or, along y,

a = ∫[0,√(7/2)] (2y)-(y/2) dy
+ ∫[0,√(7/2)] (7/y)-(y/2) dy