SETTING UP AN AREA INTEGRAL

Problem: Find the area of the region in the first quadrant which lies below both the line x + y = 2 and the parabola y = x².

Method 1: Integration with respect to x Drag the red rectangular slice of area in the diagram below. Note that we need to use different formulas to describe the height of the slice depending on whether the top of the slice is cut off by the parabola or by the line.

Method 2: Integration with respect to y Here, we fill the region with horizontal slices of height dy. The area of a slice is the width (right-most x value minus leftmost x value) times the height (dy). Before we can actually do the integration, we need to write everything in terms of y. It is not necessary to split the integral into two separate pieces for the dy approach. Try both methods -- you should get the same area both ways.

Note: Since part of the area is a triangle, its area could be computed as half the base times the height. This provides a way to check your evaluation of one of the integrals.