Find the average value of `f(x) = 4 - x^2 ` on [0,3]

Answer

To find the average value of the function \( f(x) = 4 - x^2 \) on the interval \([0, 3]\), we will use the formula for the average value of a function over an interval \([a, b]\):

\[ \text{Average value} = \frac{1}{b-a} \int_a^b f(x) \, dx \]

### Step 1: Identify the interval and function
Here, we have:
- \( f(x) = 4 - x^2 \)
- The interval is \([0, 3]\), so \( a = 0 \) and \( b = 3 \).

### Step 2: Set up the integral
We need to calculate the integral of \( f(x) \) from \( 0 \) to \( 3 \):

\[ \int_0^3 (4 - x^2) \, dx \]

### Step 3: Compute the integral
We can split the integral into two parts:

\[ \int_0^3 (4 - x^2) \, dx = \int_0^3 4 \, dx - \int_0^3 x^2 \, dx \]

Calculating each integral separately:

1. **Integral of 4**:
\[ \int_0^3 4 \, dx = 4x \bigg|_0^3 = 4(3) - 4(0) = 12 \]

2. **Integral of \( x^2 \)**:
\[ \int_0^3 x^2 \, dx = \frac{x^3}{3} \bigg|_0^3 = \frac{3^3}{3} - \frac{0^3}{3} = \frac{27}{3} = 9 \]

Now, substituting back into the integral:

\[ \int_0^3 (4 - x^2) \, dx = 12 - 9 = 3 \]

### Step 4: Calculate the average value
Now we can find the average value using the formula:

\[ \text{Average value} = \frac{1}{b-a} \int_a^b f(x) \, dx = \frac{1}{3-0} \cdot 3 = \frac{3}{3} = 1 \]

### Final Answer
Thus, the average value of \( f(x) \) on the interval \([0, 3]\) is:

\[ \boxed{1} \]