Question

Find the area of the region described in the following exercises.The region bounded by y=2 / 1 + $$x^2$$ and y=1

Accepted Answer

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.

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

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

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Key Concepts

Definite Integrals

Finding Points of Intersection

Area Between Curves

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