Textbook Question
Find the area of the region described in the following exercises.
The region bounded by y=2−|x|and y=x^2
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9. Graphical Applications of Integrals
Area Between Curves
7:43 minutesProblem 6.2.51
Textbook Question
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
Verified step by step guidance1
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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7mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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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.
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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.
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