Find the area of the region In the XY-plane enclosed by two parabolas #y=x^2# and #y=2x-x^2#?

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Answer 1 · 2018-05-04T18:15:48

#"Area" = 1/3" units"^2#

We shall find the x coordinates of the region by setting #y = y#

#x^2 =2x-x^2#

#2x^2-2x = 0#

#x(x-1)=0#

#x = 0 and x = 1#

The graphs of the two functions should show us that the enclosed area is from 0 to 1:

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It does.

The area of the region is:

#"Area" = int_0^1 2x-x^2 - x^2 dx#

#"Area" = int_0^1 2x-2x^2 dx#

#"Area" = x^2-2/3x^3|_0^1#

#"Area" = 1^2-2/3(1)^3-(0^2-2/3(0)^3)#

#"Area" = 1/3" units"^2#