The values of a function \(f(x)\) obtained for different values of x are given below
Using Simpson's one-third rule, the value of \(\int_{0}^{1} f(x) dx\) is approximately equal to ______}
Show Hint
Simpson's 1/3rd Rule can only be applied when the number of intervals ($n$) is even.
Always double-check that the step size $h$ is correctly identified; here, the difference between consecutive $x$ values is $1/4 = 0.25$.
Step 1: Understanding the Question:
We need to approximate the value of the definite integral of a function \(f(x)\) over the interval \([0, 1]\) using Simpson's 1/3rd Rule with tabular data. Step 2: Key Formula or Approach:
Simpson's 1/3rd Rule is defined as:
\[ \int_{a}^{b} f(x) dx \approx \frac{h}{3} \left[ y_{0} + y_{n} + 4(y_{1} + y_{3} + \dots + y_{n-1}) + 2(y_{2} + y_{4} + \dots + y_{n-2}) \right] \]
where:
\(h\) is the step size (interval width), \(h = \frac{b-a}{n}\).
\(y_i\) are the function values corresponding to the grid points. Step 3: Detailed Explanation:
• Step 3.1: Identify the parameters from the table:
Number of intervals, \(n = 4\) (which is an even number, satisfying the condition for Simpson's 1/3rd rule).
Step size, \(h = \frac{1}{4} = 0.25\).
The function values are:
\(y_{0} = 0.9\) (at \(x = 0\))
\(y_{1} = 2.0\) (at \(x = 1/4\))
\(y_{2} = 1.5\) (at \(x = 1/2\))
\(y_{3} = 1.8\) (at \(x = 3/4\))
\(y_{4} = 0.4\) (at \(x = 1\))
• Step 3.2: Group terms for the formula:
Sum of extreme ordinates: \(y_{0} + y_{4} = 0.9 + 0.4 = 1.3\)
Sum of odd-indexed ordinates: \(y_{1} + y_{3} = 2.0 + 1.8 = 3.8\)
Sum of even-indexed ordinates: \(y_{2} = 1.5\)
• Step 3.3: Substitute values into the equation:
\[ \int_{0}^{1} f(x) dx \approx \frac{0.25}{3} \left[ (y_{0} + y_{4}) + 4(y_{1} + y_{3}) + 2(y_{2}) \right] \]
\[ \int_{0}^{1} f(x) dx \approx \frac{0.25}{3} \left[ 1.3 + 4(3.8) + 2(1.5) \right] \]
\[ \int_{0}^{1} f(x) dx \approx \frac{0.25}{3} \left[ 1.3 + 15.2 + 3.0 \right] \]
\[ \int_{0}^{1} f(x) dx \approx \frac{0.25}{3} \left[ 19.5 \right] \]
Calculate the final product:
\[ \int_{0}^{1} f(x) dx \approx 0.25 \times 6.5 = 1.625 \]
Rounding to the nearest given option yields 1.6. Step 4: Final Answer:
The approximate value of the integral is 1.6.
