# Calculate the area bounded by the curves # y=x^2-4x+5# and #y=-2x+8 #?

##### 1 Answer

Jan 2, 2018

# 32/3 #

#### Explanation:

graph{(y-(x^2-4x+5))(y-(-2x+8))=0 [-5, 5, -5, 12]}

First we solve the simultaneous equations

# { (y=x^2-4x+5), (y=-2x+8) :} #

Which requires that:

# x^2-4x+5 = -2x+8 #

# :. x^2 -2x-3 = 0#

# :. (x+1)(x-3)= 0 #

# :. x=-1,=3#

Hence, the area sought is given by:

# A = int_(-1)^(3) \ (-2x+8) - (x^2-4x+5) \ dx #

# \ \ = int_(-1)^(3) \ -2x+8 - x^2+4x-5 \ dx #

# \ \ = int_(-1)^(3) \ - x^2+2x+3 \ dx #

# \ \ = [ - x^3/3 + x^2 +3x ]_(-1)^(3) #

# \ \ = (-9 +9+9)-(1/3+1-3)#

# \ \ = 9-(-5/3)#

# \ \ = 32/3#