Find the area enclosed by the curves \( y = x^2 \) and \( y = 4x - x^2 \).

["# Find the Area Enclosed by the Curves ( y = x^2 ) and ( y = 4x - x^2 )", "Understanding the area enclosed by two curves is a fundamental concept in calculus. This article explores how to compute the area enclosed between the parabolas ( y = x^2 ) and ( y = 4x - x^2 ), providing step-by-step instructions and clarifying key mathematical principles.", "## Step 1: Find the Points of Intersection", "To determine the area enclosed by two curves, we first need to identify where they intersect. These points occur where ( y = x^2 ) equals ( y = 4x - x^2 ):", "[\nx^2 = 4x - x^2\n]", "Bring all terms to one side:", "[\nx^2 - (4x - x^2) = 0 \implies 2x^2 - 4x = 0\n]", "Factor the equation:", "[\n2x(x - 2) = 0\n]", "Solving this gives:", "[\nx = 0 \quad \ ext{or} \quad x = 2\n]", "Thus, the curves intersect at ( x = 0 ) and ( x = 2 ). These values define the limits of integration.", "## Step 2: Determine Which Curve is Above the Other", "Between ( x = 0 ) and ( x = 2 ), we compare ( 4x - x^2 ) and ( x^2 ) to identify which function lies above.", "At ( x = 1 ):\n- ( y = x^2 = 1 )\n- ( y = 4x - x^2 = 4(1) - 1^2 = 3 )", "Since ( 3 > 1 ), the curve ( y = 4x - x^2 ) lies above ( y = x^2 ) on ( [0, 2] ). Therefore, the area ( A ) is given by the integral of the difference:", "[\nA = \int_0^2 \left[(4x - x^2) - x^2\right],dx = \int_0^2 \left(4x - 2x^2\right),dx\n]", "## Step 3: Evaluate the Integral", "Compute the definite integral:", "[\nA = \int_0^2 (4x - 2x^2),dx\n]", "Find the antiderivative:", "[\n\int (4x - 2x^2),dx = 2x^2 - \frac{2}{3}x^3 + C\n]", "Now evaluate from 0 to 2:", "[\nA = \left[2x^2 - \frac{2}{3}x^3\right]_0^2 = \left(2(2)^2 - \frac{2}{3}(2)^3\right) - \left(0\right)\n]", "[\n= \left(2 \cdot 4 - \frac{2}{3} \cdot 8\right) = 8 - \frac{16}{3} = \frac{24}{3} - \frac{16}{3} = \frac{8}{3}\n]", "## Step 4: Final Answer", "The area enclosed by the curves ( y = x^2 ) and ( y = 4x - x^2 ) is:", "[\n\boxed{\frac{8}{3} \ ext{ square units}}\n]", "This calculation illustrates the powerful technique of using definite integrals to find geometric regions bounded by polynomial curves—crucial for applications in physics, engineering, and data analysis.", "---", "Keywords: area between curves, calculus integral, find area enclosed by ( y = x^2 ), ( y = 4x - x^2 ), definite integral example, intersecting functions, mathematical problem solving.", "Meta Description: Learn how to find the area between the parabolas ( y = x^2 ) and ( y = 4x - x^2 ) using calculus. Step-by-step solution with integration and evaluation."]