Integral Of 2xdx From 10 To 13 Answer Shocks Students
Integral of 2x dx from 10 to 13: A Fast Method for quick results
The definite integral ∫₁₀¹³ 2x dx evaluates to 2.5 x (13² - 10²) or, more directly, to 13² - 10², which equals 169 - 100 = 69. This is the exact value, and the quick method relies on recognizing the antiderivative and applying the Fundamental Theorem of Calculus. In practice, the result is 69, obtained by computing the antiderivative F(x) = x² and evaluating F - F.
Derivation in a compact form
We start with the antiderivative of 2x, which is x². Applying the limits 10 and 13 gives:
- Compute F = 13² = 169.
- Compute F = 10² = 100.
- Subtract: 169 - 100 = 69.
Therefore, ∫₁₀¹³ 2x dx = 69. For quick checks, one can use a geometric interpretation: the integral of 2x over an interval corresponds to the area under the line y = 2x, which, when integrated over , aligns with the algebraic result above.
Why this method is reliable for Marist education leaders
Educational leadership often relies on precise, efficient problem-solving. The calculus method shown here mirrors how leaders approach policy analyses: identify the core relation, apply a straightforward calculation, and verify with a quick check. This mirrors the disciplined thinking we encourage in Marist schools across Latin America, where rigorous mathematics supports robust decision-making and data-informed governance.
Practical applications in school data analysis
While the problem is mathematical, the mindset translates to real-world school leadership tasks. Consider these data-analysis applications where a fast integral-like approach offers value:
- Estimating cumulative growth in enrollment metrics over a defined period by integrating a rate function.
- Calculating total resource usage when a daily consumption rate y(t) is modeled as a linear function.
- Assessing area-based funding models where budgets grow linearly with time or student numbers.
Illustrative example table
| Quantity | Definition | Calculation |
|---|---|---|
| Antiderivative | For f(x) = 2x, the antiderivative is F(x) = x² | |
| Limits | From x = 10 to x = 13 | |
| Result | F - F = 169 - 100 = 69 |
Common questions
FAQ
For administrators seeking a concise, reliable reference, the headline result for ∫₁₀¹³ 2x dx remains 69, derived from the elegant simplicity of x² as the antiderivative and the Fundamental Theorem of Calculus.
Key concerns and solutions for Integral Of 2xdx From 10 To 13 Answer Shocks Students
What is the integral of 2x from a to b?
The integral ∫ₐᵇ 2x dx equals b² - a², since the antiderivative of 2x is x². This compact formula makes quick checks straightforward and scalable to any interval [a, b].
How can I verify the result quickly?
Use the Fundamental Theorem of Calculus: evaluate the antiderivative at the upper and lower limits and subtract. For 2x, this is simply b² - a², which provides a fast confirmation.
Why choose a rapid method in education audits?
In audits and strategic planning, speed and accuracy are essential. A direct antiderivative approach minimizes errors and speeds up scenario analyses, aligning with the disciplined, values-driven style of Marist educational leadership.
