Final answer:
The area of the region between the curves y = x² and y = 2x is -2/3 square units.
Step-by-step explanation:
To find the area of the region between the curves y = x² and y = 2x, we need to determine the points at which the curves intersect. By setting the two equations equal to each other, we get:
x² = 2x
By subtracting 2x from both sides and rearranging, we get:
x² - 2x = 0
Factoring out x, we get:
x(x - 2) = 0
This means that x = 0 or x = 2. These are the points where the curves intersect.
To find the area between the curves, we integrate the difference between the two equations over the interval [0, 2]. The area can be calculated as:
Area = ∫(2x - x²) dx, from 0 to 2
By integrating, we get:
Area = [(x²/2) - (x³/3)] evaluated from 0 to 2
Plugging in the values, we get:
Area = [(2²/2) - (2³/3)] - [(0²/2) - (0³/3)]
Area = (2 - 8/3) - 0
Area = 6/3 - 8/3
Area = -2/3
Therefore, the area of the region between the curves y = x² and y = 2x is -2/3 square units.