The square root of the sum of the square of a variable 'x' and the square of a constant 'a'. This expression is commonly used in the context of trigonometric substitution, a technique employed in integral calculus to simplify the evaluation of certain types of integrals.
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The expression $\sqrt{x^2 + a^2}$ represents the length of the hypotenuse of a right triangle, where 'x' is the length of one side and 'a' is the length of the other side.
In the context of trigonometric substitution, this expression is often used to simplify integrals involving $\sqrt{a^2 - x^2}$ or $\sqrt{x^2 - a^2}$.
The trigonometric substitution $x = a\sin\theta$ transforms the integral $\int\sqrt{a^2 - x^2}\,dx$ into $\int a\cos\theta\,d\theta$, which is easier to evaluate.
The trigonometric substitution $x = a\cosh\theta$ transforms the integral $\int\sqrt{x^2 - a^2}\,dx$ into $\int a\sinh\theta\,d\theta$, which is also easier to evaluate.
The expression $\sqrt{x^2 + a^2}$ also arises in the context of polar coordinates, where it represents the distance of a point from the origin.
Review Questions
Explain how the expression $\sqrt{x^2 + a^2}$ is used in the context of trigonometric substitution.
In the context of trigonometric substitution, the expression $\sqrt{x^2 + a^2}$ is used to simplify integrals involving $\sqrt{a^2 - x^2}$ or $\sqrt{x^2 - a^2}$. By making a trigonometric substitution, such as $x = a\sin\theta$ or $x = a\cosh\theta$, the original integral can be transformed into a more manageable form that is easier to evaluate. The $\sqrt{x^2 + a^2}$ expression represents the length of the hypotenuse of a right triangle, which is a key aspect of the trigonometric substitution technique.
Describe the relationship between the expression $\sqrt{x^2 + a^2}$ and polar coordinates.
In the context of polar coordinates, the expression $\sqrt{x^2 + a^2}$ represents the distance of a point from the origin. The polar coordinate system specifies the location of a point in a plane by a distance from a reference point (the origin) and an angle from a reference direction. The distance from the origin is given by the square root of the sum of the squares of the x and y coordinates, which is precisely the expression $\sqrt{x^2 + a^2}$. This connection between the expression and polar coordinates is important in understanding the various applications of $\sqrt{x^2 + a^2}$ in mathematics.
Analyze how the expression $\sqrt{x^2 + a^2}$ is related to hyperbolic functions and their applications.
The expression $\sqrt{x^2 + a^2}$ is also related to hyperbolic functions, which are closely tied to the exponential function. In certain integral calculus problems, the trigonometric substitution $x = a\cosh\theta$ can be used to transform an integral involving $\sqrt{x^2 - a^2}$ into one involving hyperbolic functions, such as $\sinh\theta$ and $\cosh\theta$. This transformation can simplify the evaluation of the original integral, just as the use of trigonometric substitutions involving $\sin\theta$ and $\cos\theta$ can simplify integrals with $\sqrt{a^2 - x^2}$. Understanding the connections between $\sqrt{x^2 + a^2}$, trigonometric substitution, and hyperbolic functions is crucial for mastering advanced integral calculus techniques.
Related terms
Trigonometric Substitution: A method in integral calculus where a change of variable is made using a trigonometric function to transform the original integral into a more manageable form.
A coordinate system that specifies the location of a point in a plane by a distance from a reference point and an angle from a reference direction.
Hyperbolic Functions: A set of mathematical functions that are closely related to the circular trigonometric functions, but are defined in terms of the exponential function.