Find the area between the curves y = x^3 and y = x from x = 0 to x = 1.

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Q: Find the area between the curves y = x^3 and y = x from x = 0 to x = 1.

Solution: The area between the curves is given by ∫(from 0 to 1) (x - x^3) dx = [x^2/2 - x^4/4] from 0 to 1 = (1/2 - 1/4) = 1/4.

Steps: 8

Step 1: Identify the curves. We have two curves: y = x^3 and y = x.
Step 2: Determine the points of intersection. Set x^3 = x to find where the curves meet. This gives us x(x^2 - 1) = 0, so x = 0 and x = 1.
Step 3: Set up the integral to find the area between the curves. The area is given by the integral of the top curve minus the bottom curve from x = 0 to x = 1. Here, y = x is the top curve and y = x^3 is the bottom curve.
Step 4: Write the integral: ∫(from 0 to 1) (x - x^3) dx.
Step 5: Calculate the integral. First, find the antiderivative: ∫(x - x^3) dx = (x^2/2 - x^4/4).
Step 6: Evaluate the antiderivative from 0 to 1: [(1^2/2 - 1^4/4) - (0^2/2 - 0^4/4)].
Step 7: Simplify the result: (1/2 - 1/4) = 1/4.
Step 8: Conclude that the area between the curves from x = 0 to x = 1 is 1/4.