Using the data `e = 2.72, e^2 = 7.39, e^3 = 20.09,e^4 = 54.60`, the value of the definite integral `int_0^4 e^x dx `by Simpson's rule is

Options

A. 53.6
B. 53.7
C. 53.873
D. none of these

Answer

To solve the definite integral \( \int_0^4 e^x \, dx \) using Simpson's rule, we will follow these steps:

### Step 1: Identify the intervals and function values
We are given the values of \( e^x \) at specific points:
- \( e^0 = 2.72 \) (this is \( y_0 \))
- \( e^2 = 7.39 \) (this is \( y_2 \))
- \( e^3 = 20.09 \) (this is \( y_3 \))
- \( e^4 = 54.60 \) (this is \( y_4 \))

We need to find \( y_1 \) which is \( e^1 \). Since we do not have this value directly, we can approximate it using the given values. However, for the purpose of Simpson's rule, we can assume \( y_1 \) as \( e^1 \approx 2.72 \) (for simplicity, but ideally, we should calculate it).

### Step 2: Apply Simpson's Rule
Simpson's rule states that:
\[ \int_a^b f(x) \, dx \approx \frac{h}{3} \left( y_0 + 4y_1 + 2y_2 + 4y_3 + y_4 \right) \]
where \( h \) is the width of the intervals. Here, \( a = 0 \), \( b = 4 \), and we have 4 intervals (0 to 1, 1 to 2, 2 to 3, 3 to 4).

The width of each interval \( h = \frac{b-a}{n} = \frac{4-0}{4} = 1 \).

### Step 3: Substitute the values into Simpson's formula
Using the values we have:
- \( y_0 = 2.72 \)
- \( y_1 \approx 2.72 \) (assuming)
- \( y_2 = 7.39 \)
- \( y_3 = 20.09 \)
- \( y_4 = 54.60 \)

Substituting these values into the formula:
\[ \int_0^4 e^x \, dx \approx \frac{1}{3} \left( 2.72 + 4(2.72) + 2(7.39) + 4(20.09) + 54.60 \right) \]

### Step 4: Calculate the expression
Calculating each term:
- \( 4(2.72) = 10.88 \)
- \( 2(7.39) = 14.78 \)
- \( 4(20.09) = 80.36 \)

Now, adding these values:
\[ 2.72 + 10.88 + 14.78 + 80.36 + 54.60 = 163.34 \]

### Step 5: Final calculation
Now, multiply by \( \frac{1}{3} \):
\[ \int_0^4 e^x \, dx \approx \frac{1}{3} \times 163.34 \approx 54.45 \]

### Step 6: Conclusion
Thus, the value of the definite integral \( \int_0^4 e^x \, dx \) by Simpson's rule is approximately \( 54.45 \).