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x^2 = 4x - x^2.
2x^2 - 4x = 0.
Factoring out \( 2x \):
2x(x - 2) = 0.
Thus, the points of intersection are \( x = 0 \) and \( x = 2 \).
The area \( A \) between the curves from \( x = 0 \) to \( x = 2 \) is given by:
A = \int_0^2 \left((4x - x^2) - x^2\right) \, dx = \int_0^2 (4x - 2x^2) \, dx.
Evaluating the integral:
A = \left[ 2x^2 - \frac{2}{3}x^3 \right]_0^2 = \left( 2(2)^2 - \frac{2}{3}(2)^3 \right) - \left( 2(0)^2 - \frac{2}{3}(0)^3 \right).
Calculating each term: