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Find the area of the region. $$y=\frac{5}{x^{2}+1}$$

   Find the area of the region.
$$y=\frac{5}{x^{2}+1}$$

Calculus of a Single Variable

Calculus of a Single Variable

Ron Larson, Bruce… 9th Edition

Chapter 8, Problem 76

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Solved by Expert Darshan Maheshwari

11/17/2020

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Therefore, we can find the area of the region to the right of the y-axis and then double it to get the total area. Show more…

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Find the area of the region. $$y=\frac{5}{x^{2}+1}$$

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Key Concepts

Symmetry in Integration

Symmetry properties of functions, such as even or odd behavior, can greatly simplify the process of integration. For even functions, integrating over symmetric intervals about zero allows one to compute the integral over just half the domain and then double the result, reducing computational effort and complexity.

Improper Integrals

Improper integrals extend the idea of definite integrals to cases where one or both of the limits of integration are infinite or where the integrand has discontinuities. They require careful handling using limits to determine whether the integral converges to a finite value, which is crucial for computing areas over unbounded regions.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arctan(x), frequently appear in integration when dealing with rational functions like 1/(x^2+1). Recognizing that the derivative of arctan(x) is 1/(1+x^2) permits the direct computation of integrals that involve these forms, making them pivotal in many area calculations.

Area Under a Curve

This concept involves computing the definite integral of a function over a given interval to determine the accumulated area between the graph of the function and the x-axis. It forms the basis for quantifying the space or 'area' trapped by curves and set boundaries in the coordinate plane.

Definite Integration

Definite integration is the process of evaluating an integral with specified upper and lower limits. This technique is essential for calculating exact areas, total accumulations, and net changes, and it is widely used in many applications within calculus.

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