A region in the xy-plane is bounded by the curves y=4x-x^2 and y=2x-3 a. find the points of intersection. b. sketch the region bounded by the curves and label intersections and shade the bounded region. c. find the area of the region.?

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Answer 1 · 2018-05-07T06:05:08

Please see the explanation below

The points of intersection is obtained by solving the equation

#4x-x^2=2x-3#

#<=>#, #x^2-2x-3=0#

#<=>#, #(x-3)(x+1)=0#

#<=>#, #{(x-3=0),(x+1=0):}#

#<=>#, #{(x=3),(x=-1):}#

The points are #(-1,-5)# and #(3,3)#

graph{(y-4x+x^2)(y-2x+3)=0 [-8.97, 16.34, -7.14, 5.52]}

The area is

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

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

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

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

#=9+5/3#

#=32/3u^2#

#=10.67u^2#