Find the area of the region described in the following exercises.

The region bounded by y=e^x, y=e^−2x, and x=ln 4

검증된 단계별 안내

Identify the curves and boundaries that enclose the region: the curves are \( y = e^{x} \), \( y = e^{-2x} \), and the vertical line \( x = \ln 4 \).

Find the points of intersection between the two curves \( y = e^{x} \) and \( y = e^{-2x} \) by setting them equal: \( e^{x} = e^{-2x} \). Solve for \( x \) to determine the left boundary of the region.

Determine which curve is on top and which is on the bottom between the intersection point and \( x = \ln 4 \) by comparing \( e^{x} \) and \( e^{-2x} \) in that interval.

Set up the integral for the area of the region as the integral from the left intersection point to \( x = \ln 4 \) of the difference between the top curve and the bottom curve: \[ \text{Area} = \int_{x=a}^{\ln 4} \left( e^{x} - e^{-2x} \right) \, dx \], where \( a \) is the intersection point found earlier.

Evaluate the integral by integrating each term separately: \( \int e^{x} \, dx \) and \( \int e^{-2x} \, dx \), then apply the limits of integration to find the area.

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주요 개념

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

Understanding the Region Bounded by Curves

To find the area between curves, identify the region enclosed by the given functions and boundaries. This involves determining the points of intersection and the limits of integration, ensuring the correct curves are used as upper and lower bounds within the specified interval.

Definite Integration for Area Calculation

The area between two curves y = f(x) and y = g(x) over an interval [a, b] is found by integrating the difference f(x) - g(x) with respect to x. Definite integrals compute the exact area under the curve between the limits, accounting for the vertical distance between the functions.

Properties of Exponential Functions

Exponential functions like y = e^x and y = e^{-2x} have distinct growth and decay behaviors. Understanding their shapes and intersections helps in setting up the integral correctly, especially when determining which function is on top or bottom within the interval.

Find the area of the region described in the following exercises.The region bounded by y=e^x, y=e^−2x, and x=ln 4