Help with curves please!

Question

Find the area of the region bounded by the graphs of y = x² - 4x and y = x - 4.

Options:

1)	-4.500
2)	4.500
3)	2.333
4)	None of these

Answer 1

Hi Cheyanna,

First: find the intersect between given line and curve y = x² - 4x and y = x - 4.

x - 4 = x^2 -4x

x - 4 -x^2 + 4x = 0

-x^2 +5x -4 = 0

-( x - 1) (x - 4 ) = 0

x - 1 =0 ----> x = 1 and x - 4 = 0 -------> x = 4 these numbers are the bounded of integral

Area = ∫(-x^2 +5x -4) dx = - ∫(x^2 -5x +4) dx it is easy to take negative of the expression

Area = - ( 1/3 x^3 - 5/2 x^2 +4x ) from x =1 to x = 4

Area = - ( 64/3 - 5/2 *16 +16 ) + ( 1/3 -5/2 +4) = -63/3 +40 -16 -2.5 + 4 = -21 +40 -16 -2.5 +4 = 4.5

I hope it is useful,

Minoo