https://file.aiquickdraw.com/mathbot/file/17265426468032iczrmu6.jpg

Let's analyze the mathematical problem from the image. The question provides the functions \( f(x) \) and \( g(x) \), and asks us to find the value of an integral. Here's the breakdown:

### Given:
- \( f(x) = 5x^2 + 2016x^2 \) for \( 0 < x \leq a \)
- \( g(x) = f(x + 2a) = 5x^2 + 2016x^2 \) for \( -a < x \leq a \)
- We are asked to compute the integral \( \int_0^a (f(x) + g(x)) \, dx \)

The multiple-choice options are:
- a. \( 4b \)
- b. \( 5b \)
- c. \( 6b \)
- d. \( 8b \)
- e. \( 9b \)

### Approach:
We need to integrate the sum \( f(x) + g(x) \) over the interval \( [0, a] \). I will compute the sum of the functions and then proceed with the integration.

Let me calculate that for you.The result of the integral \( \int_0^a (f(x) + g(x)) \, dx \) is:

\[
\frac{40420}{3} a^3
\]

This result corresponds to the variable \( b \) being \( b = \frac{40420}{3} a^3 \), matching the expression provided in the multiple-choice format.

Would you like further clarification on this result or its derivation?

Here are some related questions:
1. What is the method to integrate a polynomial function?
2. How do we handle functions like \( f(x + a) \) during integration?
3. How can you interpret the coefficients in polynomial integrals?
4. What are the properties of definite integrals for symmetric functions?
5. How can substitutions help in evaluating more complex integrals?

**Tip:** For polynomial functions, integrating each term individually makes the process much simpler.

Diketahui fungsi f dan g dengan f(x) = f(x+a) f(x) = 5x^2 + 2016x^2 untuk 0 < x ≤ a, dengan g(x) = g(x + 2a), g(x) = 5x^2 + 2016x^2 untuk −a < x ≤ a dan ∫g₀^a f(x) dx =. Nilai dari ∫₀^a (f(x) + g(x)) dx adalah...

Learn how to calculate the definite integral of the sum of polynomial functions f(x) and g(x) in the given range. This problem involves integral calculus concepts and is suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation