A = \int_0^2 \left((4x - x^2) - x^2\right) \, dx = \int_0^2 (4x - 2x^2) \, dx.

["Title: Mastering定积分 A = ∫₀² ((4x - x²) - x²) dx: Step-by-Step Evaluation & Calculation", "---", "Introduction: Solving the Mysterious Definite Integral A = ∫₀² ((4x - x²) - x²) dx", "Calculus enthusiasts and students alike often encounter integrals involving polynomial functions, but not all integrals are straightforward. One such integral simplifies elegantly but demands careful attention:", "[\nA = \int_0^2 \left((4x - x^2) - x^2\right) , dx = \int_0^2 (4x - 2x^2) , dx\n]", "But why does this expression appear simpler? How do we compute it correctly? And what does it reveal about fundamental calculus concepts?", "This comprehensive guide breaks down the evaluation of this definite integral step-by-step, clears up common confusion, and explains its significance in integration techniques and applications.", "---", "### Step 1: Simplify the Integrand — Why Combine Like Terms?", "The expression inside the integral starts as:", "[\n(4x - x^2) - x^2\n]", "Simplifying the terms:", "[\n(4x - x^2 - x^2) = 4x - 2x^2\n]", "So,", "[\nA = \int_0^2 (4x - 2x^2) , dx\n]", "Why simplify first?\nThough often straightforward, simplifying the integrand preserves clarity and reduces computational error. It reveals the exact function being integrated and prepares you for efficient antiderivative techniques.", "---", "### Step 2: Compute the Antiderivative", "Now integrate term-by-term over the interval ([0, 2]):", "[\nA = \int_0^2 (4x - 2x^2) , dx = \int_0^2 4x , dx - \int_0^2 2x^2 , dx\n]", "Calculate each integral:", "1. (\int_0^2 4x , dx = 4 \cdot \frac{x^2}{2} \Big|_0^2 = 2x^2 \Big|_0^2 = 2(4) - 0 = 8)", "2. (\int_0^2 2x^2 , dx = 2 \cdot \frac{x^3}{3} \Big|_0^2 = \frac{2}{3}x^3 \Big|_0^2 = \frac{2}{3}(8) - 0 = \frac{16}{3})", "Now combine:", "[\nA = 8 - \frac{16}{3} = \frac{24}{3} - \frac{16}{3} = \frac{8}{3}\n]", "---", "### Step 3: Verification — Evaluating Definite Integral Directly", "Alternatively, integrate first and evaluate at bounds:", "[\nA = \int_0^2 (4x - 2x^2) , dx = \left[ 2x^2 - \frac{2}{3}x^3 \right]_0^2\n]", "Plug in (x = 2):", "[\n2(2)^2 - \frac{2}{3}(2)^3 = 2(4) - \frac{2}{3}(8) = 8 - \frac{16}{3} = \frac{24 - 16}{3} = \frac{8}{3}\n]", "Setting (x = 0) yields 0. Confirms the same result.", "---", "### Step 4: Interpretation — What Does This Integral Represent?", "The integral computes the net area between the curve (y = 4x - 2x^2) and the (x)-axis from (x = 0) to (x = 2). The function opens downward (quadratic with negative leading coefficient), crosses the (x)-axis at (x = 0) and (x = 2), forming a region above the axis. The result (\frac{8}{3}) quantifies this area.", "---", "### Step 5: Beyond the Calculation — Techniques & Tips", "- Simplify Before Integrating: Combining terms reduces work and minimizes mistakes.\n- Use Linearity of Integrals: Break complex expressions into simpler terms.\n- Apply Fundamental Theorem: Evaluate antiderivative at bounds for efficiency.\n- Check Vertical Intercepts: Always locate where functions cross the axis for sign analysis (important in area vs. definite integral contexts).", "---", "### Conclusion: The Power of Proper Integration", "The expression (A = \int_0^2 \left((4x - x^2) - x^2\right) , dx) simplifies naturally but stands as a robust example of effective integration strategy. By carefully combining terms, computing antiderivatives, and evaluating definite bounds, we find:", "[\n\boxed{A = \frac{8}{3}}\n]", "This problem is more than a numerical answer — it embodies clarity, precision, and the logical flow central to mastering calculus. Whether preparing for exams or real-world applications in physics, engineering, or economics, such skills form essential foundations.", "---", "Keywords: definite integral, A = ∫₀² (4x - x²) - x² dx, ∫₀² (4x - 2x²) dx, calculus examples, integration techniques, net area under curve, evaluate definite integral, antiderivative, step-by-step integration.", "---", "Meta Description:\nLearn how to compute and interpret the definite integral (A = \int_0^2 \left((4x - x^2) - x^2\right) dx = \int_0^2 (4x - 2x^2) dx), step-by-step. Discover simplification, antiderivative computation, and real-world significance in calculus. Perfect for students and learners.", "---", "Read More:\n- How to Determine the Area Under a Curve with Integrals\n- Step-by-Step Guide to Definite Integrals\n- Essential Techniques for Evaluating Polynomial Integrals"]