Question

$$ y=e^{x / 2}+e^{-x / 2}, \quad y=0, \quad x=-1, \quad x=2 $$

   $$
y=e^{x / 2}+e^{-x / 2}, \quad y=0, \quad x=-1, \quad x=2
$$

Calculus

Ron Larson, Robert… 8th Edition

Chapter 7, Problem 36

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Solved by Expert Nick Johnson

12/18/2021

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$$ y=e^{x / 2}+e^{-x / 2}, \quad y=0, \quad x=-1, \quad x=2 $$

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Key Concepts

Exponential Functions

Exponential functions involve expressions in which the variable appears in the exponent. They are characterized by rapid growth or decay and have unique properties such as a constant relative rate of change. In many problems, understanding the behavior of exponential functions is crucial for analyzing rates, modeling real-world phenomena, and transforming expressions for further simplification.

Hyperbolic Functions

Hyperbolic functions, such as the hyperbolic cosine defined by cosh(x) = (e^x + e^(-x))/2, are analogues of the circular trigonometric functions but based on hyperbolas rather than circles. They naturally appear when dealing with expressions that combine an exponential function and its reciprocal. These functions have properties and identities that can simplify the manipulation and integration of sums like e^(x/2) + e^(-x/2).

Graphical Analysis and Region Boundaries

Graphical analysis involves understanding and interpreting curves, lines, and regions in the coordinate plane. When a problem specifies curves and boundary conditions like a function, a horizontal line, and vertical lines at fixed x-values, the key concept is to identify the enclosed region. This includes recognizing how the function behaves relative to the horizontal axis and how the vertical lines serve as limits, which is critical for setting up integrals or understanding the geometry of the region.

Definite Integration

Definite integration is a fundamental concept in calculus used to calculate the area under a curve between given limits. When boundaries are provided by vertical lines and the x-axis (or another constant function), this process involves integrating the function over the interval defined by the limits. Understanding how to set up and evaluate these integrals is essential for solving area problems as well as for applications in physics, engineering, and other fields.

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