73. Find the area between the curves y=ln(x) and y=ln(2x) from x=1 to x=5.

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Identify the two functions given: \( y = \ln(x) \) and \( y = \ln(2x) \). We want to find the area between these curves from \( x = 1 \) to \( x = 5 \).

Determine which function is on top and which is on the bottom in the interval \( [1, 5] \). Since \( \ln(2x) = \ln(2) + \ln(x) \), it is always greater than \( \ln(x) \) for \( x > 0 \). So, \( y = \ln(2x) \) is the upper curve and \( y = \ln(x) \) is the lower curve.

Set up the integral for the area between the curves as \( \int_1^5 [\ln(2x) - \ln(x)] \, dx \).

Simplify the integrand using logarithm properties: \( \ln(2x) - \ln(x) = \ln\left(\frac{2x}{x}\right) = \ln(2) \). So the integral becomes \( \int_1^5 \ln(2) \, dx \).

Evaluate the integral by integrating the constant \( \ln(2) \) over \( [1, 5] \), which is \( \ln(2) \times (5 - 1) \). This gives the area between the curves.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Definite Integrals and Area Between Curves

The area between two curves over an interval is found by integrating the difference of their functions. Specifically, the integral of the upper function minus the lower function from the lower to upper bounds gives the enclosed area.

Properties of Logarithmic Functions

Understanding logarithmic functions, such as ln(x) and ln(2x), is essential. Using log properties like ln(2x) = ln(2) + ln(x) helps simplify the integrand and makes the integration process more straightforward.

Evaluating Definite Integrals Involving Logarithms

Integrating logarithmic functions often requires integration by parts or recognizing standard integral forms. Evaluating the definite integral involves substituting the limits after integration to find the exact area.