Find the area between the curves y = x and y = x^2 from x = 0 to x = 1.

Find the area between the curves y = x and y = x^2 from x = 0 to x = 1.

Step 1: Identify the curves. We have two curves: y = x (a straight line) and y = x^2 (a parabola).
Step 2: Determine the points of intersection. Set x = x^2 to find where the curves meet. This gives us x(x - 1) = 0, so the points of intersection are at x = 0 and x = 1.
Step 3: Set up the integral to find the area between the curves. The area A is given by the integral from 0 to 1 of the top curve minus the bottom curve. Here, y = x is above y = x^2 in this interval.
Step 4: Write the integral: A = ∫(from 0 to 1) (x - x^2) dx.
Step 5: Calculate the integral. First, find the antiderivative of (x - x^2), which is (x^2/2 - x^3/3).
Step 6: Evaluate the antiderivative from 0 to 1. Plug in 1: (1^2/2 - 1^3/3) = (1/2 - 1/3).
Step 7: Simplify the result. Find a common denominator for 1/2 and 1/3, which is 6. So, (1/2 = 3/6) and (1/3 = 2/6). Thus, (3/6 - 2/6) = 1/6.
Step 8: Conclude that the area between the curves from x = 0 to x = 1 is 1/6.

Area between curves – Calculating the area between two functions by integrating the difference of the functions over a specified interval.
Definite integrals – Using definite integrals to find the area under a curve between two points.
Function intersection – Understanding where two functions intersect to determine the limits of integration.