Question

The approximate value of $\int₁⁵x²dx$ using trapezoidal rule with $n=4$ is

• A. 41
• B. 41.5
• C. 41.75
• D. 42

Hint:

The approximate value of $\int1xdx$ using trapezoidal rule with $n=4$ is

Answer 1

The Correct Option is D

To approximate the integral \(\int_1^5 x^2 \, dx\) using the trapezoidal rule with \(n=4\), we can follow these steps:

1. Calculate the width of each sub-interval (\(h\)) using the formula: \( h = \frac{b-a}{n} = \frac{5-1}{4} = 1 \).
2. Identify the interval endpoints: \( x_0 = 1, x_1 = 2, x_2 = 3, x_3 = 4, x_4 = 5 \).
3. Evaluate the function \( f(x) = x^2 \) at each endpoint:
   • \( f(x_0) = f(1) = 1^2 = 1 \)
   • \( f(x_1) = f(2) = 2^2 = 4 \)
   • \( f(x_2) = f(3) = 3^2 = 9 \)
   • \( f(x_3) = f(4) = 4^2 = 16 \)
   • \( f(x_4) = f(5) = 5^2 = 25 \)
4. Apply the trapezoidal rule formula: \( \int_1^5 x^2 \, dx \approx \frac{h}{2} \left[ f(x_0) + 2(f(x_1) + f(x_2) + f(x_3)) + f(x_4) \right] \).
5. Substitute the known values into the formula: \( \approx \frac{1}{2} \left[ 1 + 2(4 + 9 + 16) + 25 \right] \).
6. Simplify the expression:
   • Calculate the sum inside the brackets: \( 4 + 9 + 16 = 29 \).
   • Calculate: \( 2 \times 29 = 58 \).
   • Calculate the entire expression: \( \frac{1}{2} \left[ 1 + 58 + 25 \right] = \frac{1}{2} \times 84 = 42 \).

Therefore, the approximate value of the integral \(\int_1^5 x^2 \, dx\) using the trapezoidal rule with \(n=4\) is 42, which matches the given correct answer.