Area of Region Bounded by x = y² - 2 and x = y

Calculus 2 · Axiom Academy

EXAMPLE Area of Region Bounded by x = y^2 - 2 and x = y Find the area of a region bounded by two curves using horizontal strips, integrating with respect to y. Find the area of the region bounded by the curves x = y^2 - 2 and x = y . Because both curves are given as x in terms of y , this region is best handled with horizontal strips — integrating with respect to y . The shaded region lies between the line x = y (right) and the parabola x = y² − 2 (left), bounded by the intersection points (−1, −1) and (2, 2). Excellent work! You found the area using horizontal strips. Here's what this problem demonstrated: Why horizontal strips? For curves given as x = f(y) , integrating with respect to y avoids solving for y in terms of x , which can be messy or impossible. Finding intersections: Set the x-values equal and solve for y to get the limits of integration — here y = -1 and y = 2 . Right vs. left: For a horizontal strip, identify which curve has the larger x (right) and which has the smaller x (left). Here x = y is on the right. When to use it: Horizontal integration shines when curves are naturally written as x = f(y) , or when vertical strips would force you to split the region into several integrals. Choosing the right orientation for integration can dramatically simplify the work. With vertical strips here, you'd have to split the parabola into and and integrate the pieces separately.

This is the written version of the interactive lesson above. See the full Calculus 2 course.