Approximate the integral with your calculator. Round the answer to the nearest thousandth.
$\int_1^4(\ln x)^3 d x \approx[?]$
Answer (1)
Evaluate the definite integral: ∫ 1 4 ( ln x ) 3 d x ≈ 2.86610886244945 .
Round the result to the nearest thousandth.
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 and round the result 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 this result to the nearest thousandth, we get 2.866 .
Final Answer Therefore, the approximate value of the definite integral is 2.866 .
Examples
Definite integrals are used in many fields, such as physics, engineering, and economics. For example, in physics, the definite integral can be used to calculate the work done by a force over a certain distance. In engineering, it can be used to calculate the area under a curve, which can be useful for designing structures. In economics, it can be used to calculate the consumer surplus or producer surplus in a market.
