Question

Find the area of the indicated region. We suggest you graph the curves to check whether one is above the other or whether they cross, and that you use technology to check your answer.
Enclosed by y = x^2 − 4x + 1 and y = −x^2 + 4x − 5

Answer 1

Here is a sketch
http://www.wolframalpha.com/input/?i=plot+y+%3D+x%5E2+−+4x+%2B+1+and+y+%3D+−x%5E2+%2B+4x+−+5

Easy to find their intersection points:
(1, -2) and (3, -2)

effective height between x = 1 and x = 3
= (-x^2 + 4x - 5) - (x^2 - 4x + 1)
= -2x^2 +8x -6

area = ∫(-2x^2 + 8x - 6) dx from x = 1 to 3
= [(-2/3)x^3 + 4x^2 - 6x] from 1 to 3
= ( -18 + 36 - 18) - (-2/3 + 4 - 6)
= 8/3

check my arithmetic