\( [x^3 + x^2]_0^2 = (2^3 + 2^2) - (0^3 + 0^2) = 8 + 4 - 0 = 12 \).

["# Solving ( [x^3 + x^2]_0^2 ): A Step-by-Step Algebraic Breakdown", "Understanding definite interval expressions like ([x^3 + x^2]_0^2) can seem complex at first, but breaking the problem down algebraically reveals its elegant clarity. In this article, we’ll explore the full evaluation of the expression ( [x^3 + x^2]_0^2 ), simplify it step by step, and confirm how mathematical principles lead us to the correct result: 12.", "## What Does ([x^3 + x^2]_0^2) Mean?", "The notation ([f(x)]_a^b) represents evaluating the polynomial function (f(x) = x^3 + x^2) over the interval from (x = a) to (x = b) using definite integration. Specifically:", "[\n[f(x)]_a^b = \int_a^b f(x),dx\n]", "In our case, (f(x) = x^3 + x^2), (a = 0), and (b = 2). Therefore:", "[\n[ x^3 + x^2 ]_0^2 = \int_0^2 (x^3 + x^2), dx\n]", "This integral calculates the net area under the curve of (x^3 + x^2) from (x = 0) to (x = 2). Let’s compute it step by step.", "## Step-by-Step Integration", "### Step 1: Find the Antiderivative", "We begin by integrating the polynomial term-by-term.", "[\n\int (x^3 + x^2), dx = \int x^3, dx + \int x^2, dx\n]", "Apply standard power rule integration:", "[\n\int x^n, dx = \frac{x^{n+1}}{n+1}, \quad \ ext{for } n <br/>\ne -1\n]", "So:", "[\n\int x^3, dx = \frac{x^4}{4}, \quad \int x^2, dx = \frac{x^3}{3}\n]", "Therefore, the antiderivative is:", "[\nF(x) = \frac{x^4}{4} + \frac{x^3}{3}\n]", "### Step 2: Evaluate from 0 to 2", "Now, apply the Fundamental Theorem of Calculus:", "[\n\int_0^2 (x^3 + x^2), dx = F(2) - F(0)\n]", "Calculate (F(2)):", "[\nF(2) = \frac{(2)^4}{4} + \frac{(2)^3}{3} = \frac{16}{4} + \frac{8}{3} = 4 + \frac{8}{3}\n]", "Calculate (F(0)):", "[\nF(0) = \frac{(0)^4}{4} + \frac{(0)^3}{3} = 0 + 0 = 0\n]", "Subtract:", "[\nF(2) - F(0) = \left(4 + \frac{8}{3}\right) - 0 = \frac{12}{3} + \frac{8}{3} = \frac{20}{3}\n]", "Wait — this result ((\frac{20}{3})) does NOT equal 12. What has gone wrong?", "## Re-examining the Original Expression", "The initial claim — ( [x^3 + x^2]_0^2 = (2^3 + 2^2) - (0^3 + 0^2) = 8 + 4 - 0 = 12 ) — oversimplifies the integral and misapplies bounds. That formula resembles direct evaluation at endpoints without integration, but integration requires summing over an interval.", "Let’s clarify:", "If (f(x) = x^3 + x^2), then the correct definite integral setup is:", "[\n\int_0^2 f(x), dx = \int_0^2 (x^3 + x^2), dx = \frac{20}{3}, \quad \ ext{not } (f(2) - f(0))\n]", "While (f(2) = 8 + 4 = 12) and (f(0) = 0), (f(2) - f(0) = 12) does represent the definite integral only if the function is constant or approximated naively — but for non-linear functions like (x^3 + x^2), the actual area under the curve is not equal to the difference in function values.", "This discrepancy highlights a common misconception: ([f(x)]_a^b) always means integration from (a) to (b), not just (f(b) - f(a)).", "## Correct Interpretation of ( [x^3 + x^2]_0^2 )", "Thus, the proper evaluation is:", "[\n[ x^3 + x^2 ]_0^2 = \int_0^2 (x^3 + x^2), dx = \frac{20}{3}\n]", "This is approximately (6.\overline{6}), not (12). The initial equation incorrectly assumes substitution without integration.", "## Why This Matters: Understanding Definite Integrals", "This example reinforces two key mathematical principles:", "1. The Symbol ([\cdot]_a^b) Denotes Integration, not evaluation:\n [\n \int_a^b f(x),dx \quad \ ext{is the net area under } f(x) \ ext{ from } a \ ext{ to } b\n ]\n Not equal to (f(b) - f(a)), which applies only for linear functions (by the Mean Value Theorem).", "2. Polynomial Integration is Computationally Straightforward:\n For (f(x) = x^3 + x^2), use power rule:\n [\n \int_a^b (x^3 + x^2),dx = \left[\frac{x^4}{4} + \frac{x^3}{3}\right]_a^b\n ]\n Always compute antiderivative and evaluate at bounds.", "## Conclusion", "The expression ([x^3 + x^2]_0^2) correctly evaluates to (\frac{20}{3}) via definite integration — not 12. The initial equation confuses function difference with integration.", "### Key Takeaways:\n- (\int_a^b f(x),dx) ≠ (f(b) - f(a)) for non-linear (f).\n- Proper evaluation requires computing the antiderivative and applying bounds.\n- Always clarify notation: ([f]_a^b) = integral from (a) to (b).", "### If You Get 12 — What Went Wrong?\nPossibly mistaking (f(2) - f(0)) (which is 12) for the integral result. But:\n[\nf(2) - f(0) = \int_0^2 f'(x),dx \quad \ ext{(Fundamental Theorem slice), not } \int_0^2 f(x),dx\n]", "---", "Tagline: Master definite integrals by differentiating notation — definite integral notation ([f]_a^b) always means integration, not just endpoint difference.", "### Related Keywords:\n- ( \int_0^2 (x^3 + x^2),dx )\n- Definite integral calculation\n- Fundamental Theorem of Calculus\n- Polynomial integration rule\n- Why ( \int_a^b f(x)dx <br/>\ne f(b) - f(a) ) for (f(x) = x^3 + x^2)", "### Want to Try Another Integration?\nTry computing ( \int_0^2 (2x + 1)^2,dx ) using similar methods — reinforcing the difference between net areas and function differences."]