Math 165 – Quiz 10D, area between curves –
solutions
Problem 1 Find the area of the region between the curves given by y =
x2 − 1 and y = 4 − 4x.
Solution To figure out where the two given curves meet, solve
x2 − 1 = 4 − 4x
which gives x = −5 and x = 1. So the two given curves meet at (−5, 24) and
at (1, 0). A sketch shows that for x in [−5, 1], x2 − 1 < 4 − 4x, so the area is
Z 1
A =
(x2 − 1) − (4 − 4x) dx
−5
3
1
x
2
=
+ 2x − 5x
3
−5
126
= −
+ 2(25 − 1) − 5(−6) = 36.
3
Alternative: it was possible to solve this using a dy-integral. However, in
this Quiz 10D, it was not really practical or useful, so please see the solution
to Quiz 10A for an example.