Approximate the integral with your calculator. Round the answer to the nearest thousandth.
[tex]\int_1^4(\ln x)^3 d x \approx[?][/tex]
Answer (1)
The problem asks to approximate the definite integral ∫ 1 4 ( ln x ) 3 d x .
Using a calculator, the definite integral is found to be approximately 2.86610886244945 .
Rounding this result to the nearest thousandth gives 2.866 .
Therefore, the approximate value of the definite integral is 2.866 .
Explanation
Problem Setup We are asked to approximate the definite integral ∫ 1 4 ( ln x ) 3 d x to the nearest thousandth.
Calculating the Definite Integral Using a calculator, we find that the definite integral ∫ 1 4 ( ln x ) 3 d x ≈ 2.86610886244945 .
Rounding the Result Rounding the result to the nearest thousandth, we get 2.866 .
Final Answer Therefore, the approximate value of the definite integral is 2.866 .
Examples
Imagine you are designing a component in an engine, and the performance of the component is modeled by the function ( ln x ) 3 . To calculate the total work done by this component between x = 1 and x = 4 , you would need to evaluate the definite integral ∫ 1 4 ( ln x ) 3 d x . Approximating this integral allows engineers to understand the component's performance within acceptable limits.
