Question
Find the area bounded by the curves
y = x^2, y = (x - 2)^2 and the x-axis.
y = x^2, y = (x - 2)^2 and the x-axis.
Answers
oobleck
Since the curves intersect at (1,1) using vertical strips of width dx,
A = ∫[0,1] x^2 dx + ∫[1,2] (x-2)^2 dx = 1/3 + 1/3 = 2/3
or, since the region is symmetric about the line x=1,
A = 2∫[0,1] x^2 dx = 2/3
or, using horizontal strips of width dy,
A = ∫[0,1] ((2-√y) - √y) dy = 2∫[0,1] (1 - √y) dy = 2/3
A = ∫[0,1] x^2 dx + ∫[1,2] (x-2)^2 dx = 1/3 + 1/3 = 2/3
or, since the region is symmetric about the line x=1,
A = 2∫[0,1] x^2 dx = 2/3
or, using horizontal strips of width dy,
A = ∫[0,1] ((2-√y) - √y) dy = 2∫[0,1] (1 - √y) dy = 2/3