Can anyone help me with this calculus problem?

Question

Cruising around the internet for practice problems (since school starts in a while and I want to be prepared) I came upon this one problem that I have no idea how to solve.

It goes:

Find the positive value of k such that the region enclosed by y=√x, y=x/2, x=0 and x=k has an area of 2 square units.

Any help would be greatly appreciated.

Answer 1

Drawing a graph will help you see the procedure intuitively.

The graph of y=√x, (x ≥  0) is a branch of parabola with horizontal axis.

It's not difficult to see that it intersects the line y = x/2 at point A (4,2). If 0 < x < 4, √x > x/2; if x > 4, x < x/2

The area between the parabola branch and the line, and delimited by verticals x = 0 and x = k is the integral on interval (0, k) of  the function  y=|(√x) - x/2|

That integral is a function of k. Make it equal 2 and you can evaluate k.