\int_{0}^{45} (60 - (B + 15))\, dB = \int_{0}^{45} (45 - B)\, dB = \left[45B - \frac{1}{2}B^2\right]_0^{45} = 45 \cdot 45 - \frac{1}{2} \cdot 45^2 = 2025 - \frac{2025}{2} = \frac{2025}{2} = 1012.5

["Understanding the Integral ∫₀⁴⁵ (60 – (B + 15)) dB: A Step-by-Step Explanation", "When tackling integrals in calculus, breaking them down step by step can clarify not only the computation but also the underlying principles. One such integral commonly encountered is:", "[\n\int_{0}^{45} \left(60 - (B + 15)\right) , dB\n]", "This expression involves an integral over a linear function of the variable ( B ), representing a real-world scenario like cumulative growth, cost estimation, or physical accumulation processes. In this article, we will explore how to evaluate this integral from 0 to 45, verify intermediate steps, and understand the final result of 1012.5.", "---", "### Step 1: Simplify the Integrand", "Start by simplifying the expression inside the integral:", "[\n60 - (B + 15) = 60 - B - 15 = 45 - B\n]", "So the integral becomes:", "[\n\int_{0}^{45} \left(45 - B\right) , dB\n]", "---", "### Step 2: Compute the Indefinite Integral", "Next, find the antiderivative (indefinite integral) of ( 45 - B ):", "[\n\int (45 - B) , dB = 45B - \frac{1}{2}B^2 + C\n]", "This result follows directly from the basic power rule of integration:\n[\n\int B , dB = \frac{1}{2}B^2, \quad \int 45 , dB = 45B\n]", "---", "### Step 3: Evaluate the Definite Integral", "Now evaluate the definite integral from 0 to 45:", "[\n\left[45B - \frac{1}{2}B^2\right]<em 0="0">0^{45} = \left(45 \cdot 45 - \frac{1}{2} \cdot 45^2\right) - \left(45 \cdot 0 - \frac{1}{2} \cdot 0^2\right)\n]", "Simplify each term:", "- ( 45 \cdot 45 = 2025 )\n- ( \frac{1}{2} \cdot 45^2 = \frac{1}{2} \cdot 2025 = 1012.5 )", "The lower limit at ( B = 0 ) contributes nothing:", "[\n\left(2025 - 1012.5\right) - (0) = 1012.5\n]", "Thus,", "[\n\int (45 - B), dB = 1012.5}^{45\n]", "---", "### Why This Integral Matters", "This kind of integral often arises in scenarios involving linear change over time or distance. For example, if ( B ) represents a quantity increasing linearly with time over 45 units, the integral computes the cumulative total of ( (60 - (B + 15)) )—essentially a weighted or adjusted total sum. The result 1012.5 is the net contribution across the entire interval, factoring in decrement and rate of change.", "---", "### Final Answer:", "[\n\boxed{ \int_{0}^{45} (60 - (B + 15)), dB = \int_{0}^{45} (45 - B), dB = 1012.5 }\n]", "Understanding each step—simplifying expressions, integrating term-by-term, evaluating limits—empowers precise calculation and deeper comprehension of integral applications in applied mathematics and real-life modeling.", "---", "Keywords: integral calculation, definite integral, linear function integration, calculus tutorial, integral evaluation, ∫₀⁴⁵, mathematical process, applied math, solving integrals step-by-step."]