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

The region bounded by y=e^x, y=2e^−x+1, and x=0

검증된 단계별 안내

Step 1: Identify the boundaries of the region. The region is bounded by the curves y = e^x, y = 2e^(-x) + 1, and the vertical line x = 0. This means the region lies between these curves and starts at x = 0.

Step 2: Determine the points of intersection between the curves y = e^x and y = 2e^(-x) + 1. To find these points, set e^x = 2e^(-x) + 1 and solve for x. This will give the x-values where the two curves meet.

Step 3: Set up the integral to calculate the area. The area is found by integrating the difference between the upper curve (y = 2e^(-x) + 1) and the lower curve (y = e^x) over the interval determined by the points of intersection and x = 0.

Step 4: Write the integral expression for the area. The area can be expressed as: ∫[x=0 to x=intersection] [(2e^(-x) + 1) - e^x] dx. This represents the vertical distance between the curves integrated over the specified interval.

Step 5: Evaluate the integral. Break the integral into simpler parts if necessary, such as ∫[x=0 to x=intersection] 2e^(-x) dx, ∫[x=0 to x=intersection] 1 dx, and ∫[x=0 to x=intersection] e^x dx. Compute each part and combine the results to find the total area.

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주요 개념

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

Definite Integrals

Definite integrals are used to calculate the area under a curve between two points on the x-axis. In this context, the area between the curves y=e^x and y=2e^−x+1 can be found by integrating the difference of these functions over the appropriate interval. The limits of integration will be determined by the points where the curves intersect.

Curve Intersection

Finding the intersection points of the curves y=e^x and y=2e^−x+1 is essential for determining the limits of integration. This involves setting the two equations equal to each other and solving for x. The x-values at which the curves intersect will define the boundaries of the area to be calculated.

Exponential Functions

Exponential functions, such as y=e^x and y=2e^−x+1, are characterized by their rapid growth or decay. Understanding their behavior is crucial for analyzing the region they enclose. The function e^x increases without bound as x increases, while 2e^−x+1 approaches 1 as x increases, creating a bounded area between them.

Find the area of the region described in the following exercises.The region bounded by y=e^x, y=2e^−x+1, and x=0