https://file.aiquickdraw.com/mathbot/file/1728928129095exz5921s.jpg

The function given is \( f(x) = \frac{1}{4} (\pi - x)^2 \), and we are tasked with finding \( a_0 \), which likely refers to the constant term of the Fourier series expansion of \( f(x) \).

### Steps to find \( a_0 \):
In the Fourier series, the constant term \( a_0 \) is given by the average value of the function over the period. For simplicity, we assume that the interval is \( [0, 2\pi] \), which is common for Fourier series.

The formula for \( a_0 \) is:
\[
a_0 = \frac{1}{2\pi} \int_0^{2\pi} f(x) \, dx
\]
Substituting \( f(x) = \frac{1}{4} (\pi - x)^2 \):

\[
a_0 = \frac{1}{2\pi} \int_0^{2\pi} \frac{1}{4} (\pi - x)^2 \, dx
\]

We can factor out the constant \( \frac{1}{4} \) from the integral:

\[
a_0 = \frac{1}{8\pi} \int_0^{2\pi} (\pi - x)^2 \, dx
\]

### Solve the integral:
To compute \( \int_0^{2\pi} (\pi - x)^2 \, dx \), expand the square:

\[
(\pi - x)^2 = \pi^2 - 2\pi x + x^2
\]

Now, integrate each term separately:

\[
\int_0^{2\pi} \pi^2 \, dx = \pi^2 \cdot 2\pi = 2\pi^3
\]

\[
\int_0^{2\pi} (-2\pi x) \, dx = -2\pi \left[ \frac{x^2}{2} \right]_0^{2\pi} = -2\pi \cdot \frac{(2\pi)^2}{2} = -4\pi^3
\]

\[
\int_0^{2\pi} x^2 \, dx = \left[ \frac{x^3}{3} \right]_0^{2\pi} = \frac{(2\pi)^3}{3} = \frac{8\pi^3}{3}
\]

### Combine the results:
Now, sum the results of the integrals:

\[
2\pi^3 - 4\pi^3 + \frac{8\pi^3}{3}
\]
Finding a common denominator:

\[
= \frac{6\pi^3}{3} - \frac{12\pi^3}{3} + \frac{8\pi^3}{3} = \frac{2\pi^3}{3}
\]

### Final value of \( a_0 \):
Now, substitute this back into the expression for \( a_0 \):

\[
a_0 = \frac{1}{8\pi} \cdot \frac{2\pi^3}{3} = \frac{\pi^2}{12}
\]

Thus, the value of \( a_0 \) is \( \frac{\pi^2}{12} \), which corresponds to option **b)**.

Would you like more details on any step? 

Here are 5 related questions:
1. How do we calculate the higher Fourier coefficients \( a_n \)?
2. What is the physical significance of the constant term \( a_0 \) in Fourier series?
3. Could the result change if the integration bounds were different?
4. How would the Fourier series change for a periodic extension of \( f(x) \)?
5. What if the function \( f(x) \) were piecewise defined—how would the approach change?

**Tip**: Always double-check the bounds of integration in Fourier problems to ensure correct results, especially for non-standard intervals.

Find a_0 of the function f(x) = 1/4 * (π - x)^2.

Learn how to find the constant term a_0 of the function f(x) = (π - x)^2 / 4 in its Fourier series expansion. This problem involves definite integration and polynomial expansion, suitable for students in advanced calculus or Fourier analysis courses.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation