First, we need to determine the points of intersection of the curves y=x^2 and y=2x.
To find these intersections, we set both equations equal to each other:
x^2 = 2x
Rearranging, we find:
x^2 - 2x = 0
By factoring, we find solutions:
x(x - 2) = 0
Therefore, x = 0 and x = 2 are the intersection points of the given curves.
Now that we have the limits of integration, we can find the area between the curves. The area A of a region bounded by two curves y1 and y2 from x=a to x=b is given by:
A = ∫_a^b |y2 - y1| dx
Here, y1 = x^2 and y2 = 2x. Thus, we have:
A = ∫_0^2 |2x - x^2| dx
Since 2x - x^2 is non-negative in the interval [0,2], we are simply finding an area, and there are no negatives due to symmetry about the y-axis, so we can ignore the absolute values:
A = ∫_0^2 (2x - x^2) dx
This integral can be computed using the power rule, where the integral of x^n dx is (1/(n+1))*x^(n+1) from a to b.
A = 4/3 square units
And therefore, the area of the region enclosed by the curves y=x^2and y=2x is 4/3 square units.