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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 6.2.41
Chapter 6, Problem 6.2.41

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


The region bounded by y=2 / 1 + x^2 and y=1

Verified step by step guidance
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Step 1: Identify the region to be analyzed. The region is bounded by the curves y = 2 / (1 + x^2) and y = 1. To find the area, we need to determine the points of intersection between these two curves by solving the equation 2 / (1 + x^2) = 1.
Step 2: Solve the equation 2 / (1 + x^2) = 1 to find the x-values where the curves intersect. Multiply through by (1 + x^2) to eliminate the denominator, resulting in 2 = 1 + x^2. Rearrange to find x^2 = 1, and solve for x to get x = ±1.
Step 3: Set up the integral to calculate the area. The area is given by the integral of the difference between the upper curve (y = 2 / (1 + x^2)) and the lower curve (y = 1) over the interval [−1, 1]. The integral expression is: ∫[−1, 1] [(2 / (1 + x^2)) − 1] dx.
Step 4: Break the integral into simpler parts. Rewrite the integral as ∫[−1, 1] (2 / (1 + x^2)) dx − ∫[−1, 1] 1 dx. The first term involves the integral of 2 / (1 + x^2), which is a standard integral, and the second term is the integral of a constant.
Step 5: Evaluate each integral separately. The integral of 2 / (1 + x^2) dx is 2 arctan(x), and the integral of 1 dx is x. Combine these results and evaluate them at the bounds x = −1 and x = 1 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 = 2 / (1 + x^2) and y = 1 can be found by integrating the difference of these functions over the interval where they intersect.
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Definition of the Definite Integral

Finding Points of Intersection

To determine the area between two curves, it is essential to find their points of intersection. This involves setting the equations equal to each other and solving for x, which provides the limits of integration necessary for calculating the area.
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Critical Points

Area Between Curves

The area between two curves is calculated by integrating the upper function minus the lower function over the interval defined by their points of intersection. This method allows for the accurate determination of the enclosed area, which is crucial for solving the given problem.
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Finding Area Between Curves on a Given Interval
Related Practice
Textbook Question

Determine the area of the shaded region in the following figures.

Textbook Question

53–62. Choose your method Let R be the region bounded by the following curves. Use the method of your choice to find the volume of the solid generated when R is revolved about the given axis.

y = x² and y = 2−x²; about the x-axis

Textbook Question

9-34. Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis. 


y = x³−x⁸+1,y=1; about the y-axis

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Textbook Question

53–62. Choose your method Let R be the region bounded by the following curves. Use the method of your choice to find the volume of the solid generated when R is revolved about the given axis.

y = x²,y=2−x, and x=0, in the first quadrant; about the y-axis

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Textbook Question

Determine the area of the shaded region bounded by the curve x^2=y^4(1−y^3) (see figure).

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Textbook Question

Determine the area of the shaded region in the following figures.

(Hint: Find the intersection point by inspection.)