f(x) = 4x(1-x^2), x = [0,1]. what is the mean

To find the mean of the function \( f(x) = 4x(1 - x^2) \) over the interval \( [0, 1] \), we use the formula for the mean value of a function on a closed interval:

\[
\text{Mean} = \frac{1}{b-a} \int_{a}^{b} f(x) \, dx
\]

where \( a = 0 \) and \( b = 1 \). Thus, the mean becomes:

\[
\text{Mean} = \int_{0}^{1} 4x(1 - x^2) \, dx
\]

### Step 1: Expand the integrand
First, expand \( 4x(1 - x^2) \):

\[
4x(1 - x^2) = 4x - 4x^3
\]

### Step 2: Integrate term-by-term
Now, integrate each term separately:

\[
\int_{0}^{1} (4x - 4x^3) \, dx = \int_{0}^{1} 4x \, dx - \int_{0}^{1} 4x^3 \, dx
\]

Calculate these integrals:

1. \( \int_{0}^{1} 4x \, dx = 4 \left[ \frac{x^2}{2} \right]_{0}^{1} = 4 \cdot \frac{1}{2} = 2 \)

2. \( \int_{0}^{1} 4x^3 \, dx = 4 \left[ \frac{x^4}{4} \right]_{0}^{1} = 4 \cdot \frac{1}{4} = 1 \)

### Step 3: Subtract the results
The integral from 0 to 1 of \( 4x(1 - x^2) \) is:

\[
2 - 1 = 1
\]

### Step 4: Divide by the length of the interval
The length of the interval \( [0,1] \) is \( 1 \), so the mean is:

\[
\text{Mean} = \frac{1}{1} \cdot 1 = 1
\]

Therefore, the mean value of the function \( f(x) = 4x(1 - x^2) \) over the interval \( [0,1] \) is \( 1 \).

Would you like more details or have any questions?

Here are some related questions:
1. How would you find the mean for a different interval, such as \( [0,2] \)?
2. What would the mean be if the function was \( f(x) = 4x^2(1 - x^2) \)?
3. Can you find the average rate of change of \( f(x) \) over \( [0,1] \)?
4. What is the maximum value of \( f(x) \) in the interval \( [0,1] \)?
5. How would you find the median value of \( f(x) \) over the interval?

**Tip:** When integrating polynomial functions, break them down into simpler terms and integrate term-by-term for ease.

f(x) = 4x(1-x^2), x = [0,1]. What is the mean?

Discover how to calculate the mean value of the function f(x) = 4x(1 - x^2) over the interval [0,1]. This step-by-step solution includes polynomial expansion and integration, using key calculus concepts like the mean value formula. Ideal for high school calculus students in Grades 11-12.

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