Math 165 – Quiz 10B, area between curves –
solutions
Problem 1 Find the area of the region between the x-axis and the curves
given by y = 3x − x2 , x + y = 4.
Solution The two given curves meet at (2, 2). There are also the points
(0, 0) and (4, 0) where these curves intersect the x-axis. We split the given
region into the part above [0, 2] and the part above [2, 4]. The first part has
area
2
Z 2
10
3 2 1 3
2
= .
3x − x dx = x − x
2
3
3
0
0
The second part has area
Z
2
4
1
4 − x dx = 4x − x2
2
4
= 8 − 6 = 2.
2
So the total area equals 16/3.
Alternative: it was possible to solve this using a dy-integral. However, in
this Quiz 10B, it was not really practical or useful, so please see the solution
to Quiz 10A for an example.