Because the curve crosses the x-axis at , you have to divide the region into two parts: from x = 0 to x
= and from x = to x = 2. In the latter region, you’ll need to integrate y = − (2 − x^2 ) = x^2 − 2 to
adjust for the region’s being below the x-axis. Therefore, we can find the area by evaluating
(2 − x^2 ) dx + (x^2 − 2) dxIntegrating, we get
PROBLEM 3. Find the area of the region between the curve x = y^2 − 4y and the line x = y.
Answer: First, sketch the graph over the interval.