Calculate the area between the curves y = x^2 and y = 2x from x = 0 to x = 2.

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Q: Calculate the area between the curves y = x^2 and y = 2x from x = 0 to x = 2.

Solution: The area is given by the integral from 0 to 2 of (2x - x^2) dx. This evaluates to [x^2 - x^3/3] from 0 to 2 = (4 - 8/3) = 4/3.

Steps: 8

Step 1: Identify the curves. We have two equations: y = x^2 (a parabola) and y = 2x (a straight line).
Step 2: Find the points where the curves intersect. Set x^2 = 2x and solve for x. This gives us x^2 - 2x = 0, which factors to x(x - 2) = 0. So, the intersection points are x = 0 and x = 2.
Step 3: Determine which curve is on top between the intersection points. For x = 1, y = x^2 gives 1 and y = 2x gives 2. Since 2 > 1, the line y = 2x is above the parabola y = x^2 between x = 0 and x = 2.
Step 4: Set up the integral to find the area between the curves. The area A is given by the integral from 0 to 2 of (top curve - bottom curve), which is A = ∫ from 0 to 2 of (2x - x^2) dx.
Step 5: Calculate the integral. The integral of (2x - x^2) is (x^2 - (x^3)/3).
Step 6: Evaluate the integral from 0 to 2. Plug in 2: (2^2 - (2^3)/3) = (4 - 8/3).
Step 7: Simplify the result. 4 can be written as 12/3, so (12/3 - 8/3) = 4/3.
Step 8: Conclude that the area between the curves from x = 0 to x = 2 is 4/3.