Question

Find the area of the region bound by the curves y = 6x – x² and y = x² – 2x 

Answer 1

y = 6x - x², y = x² - 2x

x² - 6x  = - y

⇒  x² - 6x + 9 = - y + 9

⇒ ( x - 3 )² = - ( y - 9 )
This represents a downward parabola with vertex (3, 9)   ...(i)

y = x² - 2x

x² - 2x + 1 = y + 1

⇒  ( x - 1 )² = y + 1

⇒  ( x -1 )² = y - ( -1 )                                ...(ii)
This represents an upward parabola with vertex (1, -1)

Point of intersection is given by:

6x - x² = x² - 2x
⇒ 2x² = 8x
⇒  x² - 4x = 0
⇒  x (x - 4) = 0
x = 0, x = 4
y = 6 x 4 - 4²
= 24 - 16 = 8

Point of intersection is (0, 0) ; (4, 8)

Reqd. area = ` int_0^4 (6x - x^2) - (x^2 - 2x) dx`

= `int_0^4  8x - 2x^2 dx = [ (8x^2)/(2) - (2)/(3) x^3]_0^4`

= 4 x  4² - `(2)/(3) (4)^3 = (64)/(3)` sq. units.