advertisement

Math 165 – Quiz 10A, area between curves – solutions Problem 1 Find the√area of the region between the x-axis and the curves given by y = 2x, y = 5 − x. Solution The two given curves meet at (1, 2). There are also the points (0, 0) and (5, 0) where these curves intersect the x-axis. We split the given region into the part above [0, 1] and the part above [1, 5]. The first part has area Z 1 1 2x dx = x2 0 = 1. 0 The second part has area Z 5 √ 5 16 2 5 − x dx = − (5 − x)3/2 1 = . 3 3 1 So the total area equals 19/3. Alternative: Set up a dy-integral. Solve the given equations for x: y x = 2 x = 5 − y2 The first of these gives the lowest possible x-value for any given y, and the second gives the highest possible x-value. The possible y-values are [0, 2], also apparent from the plot of the region. So the area is Z 2 y A= (5 − y 2 ) − dy 2 0 which gives the same answer, of course. A nice alternative, since we can do the whole area with a single integral instead of two integrals.