Compute integral: ∫(400 − 10t²) dt = 400t − (10/3)t³ evaluated from 0 to 4.
["Compute the Integral ∫(400 − 10t²) dt from 0 to 4: A Step-by-Step Guide", "When solving definite integrals in calculus, one of the foundational skills is computing indefinite integrals and evaluating them between specified limits. In this article, we explore how to evaluate the integral", "[\n\int_0^4 (400 - 10t^2) , dt\n]", "using standard integration rules, demonstrating both the process and its application.", "---", "### Understanding the Integral", "The function to integrate is:", "[\nf(t) = 400 - 10t^2\n]", "We aim to compute:", "[\n\int_0^4 (400 - 10t^2) , dt\n]", "By definition, the definite integral computes the signed area under the curve ( f(t) = 400 - 10t^2 ) from ( t = 0 ) to ( t = 4 ).", "---", "### Step 1: Find the Indefinite Integral", "To integrate ( 400 - 10t^2 ), we apply basic antiderivative rules:", "[\n\int (400 - 10t^2) , dt = \int 400 , dt - 10 \int t^2 , dt\n]", "Using the power rule for integration:", "[\n\int 400 , dt = 400t + C\n]\n[\n\int t^2 , dt = \frac{t^3}{3} \quad \Rightarrow \quad -10 \int t^2 , dt = -10 \cdot \frac{t^3}{3} = -\frac{10}{3}t^3\n]", "Combining these, the indefinite integral is:", "[\n\int (400 - 10t^2) , dt = 400t - \frac{10}{3}t^3 + C\n]", "---", "### Step 2: Evaluate the Definite Integral", "Now evaluate from ( t = 0 ) to ( t = 4 ):", "[\n\int_0^4 (400 - 10t^2) , dt = \left[400t - \frac{10}{3}t^3\right]_0^4\n]", "Compute at ( t = 4 ):", "[\n400(4) - \frac{10}{3}(4)^3 = 1600 - \frac{10}{3} \cdot 64 = 1600 - \frac{640}{3}\n]", "Compute at ( t = 0 ):", "[\n400(0) - \frac{10}{3}(0)^3 = 0\n]", "Subtract:", "[\n\left(1600 - \frac{640}{3}\right) - 0 = 1600 - \frac{640}{3}\n]", "Convert 1600 to a fraction with denominator 3:", "[\n1600 = \frac{4800}{3}, \quad \ ext{so} \quad \frac{4800}{3} - \frac{640}{3} = \frac{4160}{3}\n]", "---", "### Final Result", "[\n\boxed{\int_0^4 (400 - 10t^2), dt = \frac{4160}{3}}\n]", "This value represents the net area under the parabola ( f(t) = 400 - 10t^2 ) between ( t = 0 ) and ( t = 4 ).", "---", "### Why This Integration Matters", "Integrals like this appear frequently in physics, engineering, and economics—such as computing total displacement from a velocity function, total accumulated revenue, or area in optimization problems. Mastering integration of polynomial functions builds a critical foundation for more complex calculus applications.", "---", "### Summary", "- Indefinite integral: ( \int (400 - 10t^2), dt = 400t - \frac{10}{3}t^3 + C )\n- Definite evaluation from 0 to 4 gives: ( \left[400t - \frac{10}{3}t^3\right]_0^4 = \frac{4160}{3} )\n- This area calculation is essential for solving real-world problems involving accumulation.", "---", "Keywords: compute integral, definite integral, indefinite integral, integral of 400 - 10t², area under curve, calculus tutorial, integral evaluation, antiderivative, (\int (400 - 10t^2), dt) from 0 to 4."]
