Problem

Find the area of the region bounded by y² = 2x and x – y = 4. Sketch the region.

Solution

The curves intersect when both equations hold. Solve the second equation for x:

Substitute into the first equation:

Solve for y:

The points of intersection are:

Now let's sketch their graphs.

This region looks more easily integrable with respect to y rather than x, since if we integrate with respect to y we're finding the area between two functions of y:

and

The y values over the region range from –2 to 4. So the integral is

This equals

It is possible to find the area using integrals that integrate with respect to x, but we have to break up the problem into two integrals.

Between x = 0 and x = 2, we find the area between the upper and lower branches of the quadratic curve:

Between x = 2 and x = 8, we find the area between the upper branch of the quadratic and the line: