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Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 6 - Applications of IntegrationProblem 6.2.51
Chapter 6, Problem 6.2.51

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

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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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Key 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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Related Practice
Textbook Question

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


The region bounded by y=2−|x|and y=x^2

Textbook Question

Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis.

y=0,y=lnx,y=2, and x=0; about the y-axis

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Textbook Question

Use calculus to find the volume of a tetrahedron (pyramid with four triangular faces), all of whose edges have length 4.

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Textbook Question

Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis.

y=1 / 4√1 − x^2,y=0,x=0, and x=12; about the x-axis

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Textbook Question

9–12. Consider the cylindrical tank in Example 4 that has a height of 10 m and a radius of 5 m. Recall that if the tank is full of water, then ∫₀¹⁰ 25 π ρg(15−y) dy equals the work required to pump all the water out of the tank, through an outflow pipe that is 15 m above the bottom of the tank. Revise this work integral for the following scenarios. (Do not evaluate the integrals.)


The work required to empty the tank through an outflow pipe at the top of the tank

Textbook Question

Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis.


y=|x| and y=2−x^2; about the x-axis