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

Answer 1

Emma Aldridge

#"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:

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#

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Answer 2

Evelyn Agard

To find the area of the region enclosed by two parabolas (y = x^2) and (y = 2x - x^2), you need to first determine the points where they intersect. Set the equations equal to each other and solve for (x). Then integrate the absolute difference between the functions from the leftmost intersection point to the rightmost intersection point. The integral would be: [\int_{x_1}^{x_2} |(2x - x^2) - x^2| dx].

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.