Matching Areas

Calculus Level 3

Take a point PP on the graph of y=2x2,y=2x^2, and draw a line going through PP parallel to the yy-axis. Then call the red area A:A: the area bounded by this line, the graph of y=x2,y=x^2, and the graph of y=2x2.y=2x^2.

Now, draw a line going through PP parallel to the xx-axis. Then call the blue area B:B: the area bounded by this line, the graph of y=2x2,y=2x^2, and the graph of y=f(x).y=f(x).

This function f(x)f(x) is continuous and displays the unique property that for every point PP on y=2x2,y = 2x^2, the two areas AA and BB are equal.

Find f(x)f(x) and evaluate f(12).f(12).

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