Math 165 – Quiz 10C, area between curves –
solutions
Problem 1 Find the area of the region between both coordinate axes and
the curves given by x + 2y = 12 and 4y = 5(x − 4)2 .
Solution The two given curves meet at (2, 5). 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]. Above [0, 2], the
top boundary is given by y = (12 − x)/2 = 6 − x/2 (solved the given equation
for y). So the first part has area
Z
0
2
2
x2
x
= 11.
6 − dx = 6x −
2
4 0
The second part has top boundary given by y = 5(x − 4)2 /4, so area
Z 4
4 10
5 5(x − 4)2
dx =
(x − 4)3 2 = .
4
12
3
2
So the total area equals 43/3.
Alternative: it was possible to solve this using a dy-integral. However, in
this Quiz 10C, it was not really practical or useful, so please see the solution
to Quiz 10A for an example.