Question

Value of \( \int_4^{5.2} \ln x \, dx \) using Simpson's one-third rule with interval size 0.3 is

• A. 1.83
• B. 1.60
• C. 1.51
• D. 1.06

Hint:

When using Simpson's one-third rule, ensure that the interval size \(h\) is constant. The formula provides a good approximation for smooth functions.

Answer 1

The Correct Option is A

Simpson's one-third rule is used to approximate definite integrals. The formula for Simpson's one-third rule is: \[ \int_a^b f(x) \, dx \approx \frac{h}{3} \left( f(a) + 4f\left(\frac{a+b}{2}\right) + f(b) \right) \] Where:
- \(h\) is the interval size,
- \(a\) and \(b\) are the limits of integration.
We are given the following:
- Lower limit \(a = 4\),
- Upper limit \(b = 5.2\),
- Interval size \(h = 0.3\).
Now, we calculate:
- \(f(4) = \ln(4)\),
- \(f(4.6) = \ln(4.6)\),
- \(f(5.2) = \ln(5.2)\).
We substitute these values into the Simpson's formula: \[ \int_4^{5.2} \ln x \, dx \approx \frac{0.3}{3} \left( \ln(4) + 4\ln(4.6) + \ln(5.2) \right) \] After calculating the values, we find: \[ \int_4^{5.2} \ln x \, dx \approx 1.83 \] Thus, the correct answer is (A).