Find the area of the region bounded by the curves y=x^2 & y=2x???

The curves intersect at (0,0) and (2,2)

So, you can add up horizontal strips or vertical strips. In this case, vertical strips are easier (why?)

a = ∫[0,2] (2x-x^2) dx = 4/3

To find the area of the region bounded by the curves y = x^2 and y = 2x, we need to determine the points of intersection first.

Setting the two equations equal to each other, we have x^2 = 2x. Rearranging the equation, we get x^2 - 2x = 0.

Factoring out x, we have x(x - 2) = 0.

This equation gives us two possible solutions: x = 0 and x = 2.

To determine the y-coordinates of these points of intersection, we substitute the x-values into either of the original equations.

For x = 0, y = (0)^2 = 0.

For x = 2, y = (2)^2 = 4.

So, the two points of intersection are (0, 0) and (2, 4).

To find the area between the curves, we integrate the function representing the upper curve (2x) minus the function representing the lower curve (x^2) over the interval [0, 2].

The integral for the area, A, is given by:

A = ∫(from 0 to 2) (2x - x^2) dx.

Evaluating this integral will give us the area of the region bounded by the curves y = x^2 and y = 2x.