Math Problem Statement
Average value of function f(x) = (2x^2 +1)/(e^(x/3)) over [0,3]
Solution
The average value of a continuous function over an interval is given by the formula:
For the function over the interval , we can apply this formula. Here, and . The average value becomes:
This simplifies to:
Now, to evaluate this integral, we use numerical methods because the integral doesn't have a simple elementary antiderivative.
Using numerical approximation (such as Simpson's rule or using a calculator), we find:
Thus, the average value of the function is approximately:
The average value of the function over is approximately .
Would you like further details on how to perform numerical integration?
Here are 5 related questions:
- What are common techniques for evaluating definite integrals numerically?
- How does Simpson's rule work in approximating integrals?
- What are some examples of functions with easily computable antiderivatives?
- How does the exponential decay affect the behavior of the function ?
- Can the average value formula be applied to functions with discontinuities?
Tip: For functions that don't have simple antiderivatives, numerical methods (like trapezoidal or Simpson's rule) are efficient ways to approximate definite integrals.
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Math Problem Analysis
Mathematical Concepts
Calculus
Definite Integrals
Average Value of a Function
Numerical Integration
Formulas
f_avg = (1 / (b - a)) * ∫[a to b] f(x) dx
Theorems
Fundamental Theorem of Calculus
Suitable Grade Level
Undergraduate Calculus (Grades 11-12 and beyond)
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