Let `f(0) = 1, f(1) = 2.72`, then the trapezoidal rule gives approximate value of so `int_0^1 f(x) dx `

Options

A. `3.72 `
B. `1.86 `
C. `1.72 `
D. `0.86 `

Answer

To solve the problem using the trapezoidal rule, we will follow these steps:

### Step 1: Identify the function values and the interval
We are given:
- \( f(0) = 1 \)
- \( f(1) = 2.72 \)

We need to approximate the integral \( \int_0^1 f(x) \, dx \).

### Step 2: Apply the trapezoidal rule formula
The trapezoidal rule states that:
\[ \int_a^b f(x) \, dx \approx \frac{(b - a)}{2} \left( f(a) + f(b) \right) \]
In our case, \( a = 0 \) and \( b = 1 \).

### Step 3: Calculate \( b - a \)
\[ b - a = 1 - 0 = 1 \]

### Step 4: Substitute values into the trapezoidal rule formula
Now we substitute \( a \), \( b \), \( f(0) \), and \( f(1) \) into the formula:
\[ \int_0^1 f(x) \, dx \approx \frac{1}{2} \left( f(0) + f(1) \right) \]
\[ = \frac{1}{2} \left( 1 + 2.72 \right) \]

### Step 5: Calculate \( f(0) + f(1) \)
\[ f(0) + f(1) = 1 + 2.72 = 3.72 \]

### Step 6: Calculate the final approximation
Now we can find the approximate value of the integral:
\[ \int_0^1 f(x) \, dx \approx \frac{1}{2} \times 3.72 = 1.86 \]

### Final Answer
Thus, the approximate value of \( \int_0^1 f(x) \, dx \) using the trapezoidal rule is **1.86**.

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