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.