Find the area enclosed between the curves

Solution

To find the area enclosed between the curvesy=(x−4)² and y=16−x², we first need to find the points of intersection of these two curves. We set the y values equal to each other: (x−4)² = 16−x² x² +16 – 8x = 16 – x² 2x² – 8x = 0 2x(x-4) = 0 The solutions are x=0 and x=4. These are the limits of our integration interval. Now we need to determine which function is greater in the interval [0,4]. Let's test a point within the interval, say x=2: Fory = (x−4)², when x=2, y=(2−4)² = 4 Fory = 16−x², when x=2, y=16−(2)² = 12 Since 12>4 at x=2, the curvey = 16−x²is abovey = (x−4)² in the interval [0,4]. The area enclosed between the curves is given by the integral of the difference between the upper and lower functions over the intersection interval: