Question

Evaluate the following, using either power series, a table of error functions, or asymptotic series, whichever is appropriate. $$ \frac{2}{\sqrt{\pi}} \int_{1.5}^{\infty} e^{-x^{2}} d x $$

Short answer

0.0339

Step-by-step solution

Express in terms of the error function

The integral can be expressed in terms of the error function (erf). Recall that: \[ \text{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} dt \]Therefore, we rewrite the given integral as: \[ \frac{2}{\sqrt{\pi}} \int_{1.5}^{\infty} e^{-x^{2}} dx = \text{erfc}(1.5) \]where erfc(x) is the complementary error function defined as: \[ \text{erfc}(x) = 1 - \text{erf}(x) \]

Look up the value of the complementary error function

Using a table of error functions or a calculator: \[ \text{erfc}(1.5) \approx 0.0339 \]

Solve the integral using the value of the complementary error function

Replace \( \text{erfc}(1.5) \) with its approximate value to evaluate the given integral: \[ \frac{2}{\sqrt{\pi}} \int_{1.5}^{\infty} e^{-x^{2}} dx \approx 0.0339 \]

Key concepts

error function

The error function, often denoted as \text{erf}(x), is a special function in integral calculus. It is important in probability, statistics, and partial differential equations. The error function is defined as: \[ \text{erf}(x) = \frac{2}{\text{√π}} \text{∫}_{0}^{x} e^{-t^2} dt \]
This function measures the probability that a random variable, with a normal distribution and mean zero, falls within the range \([-x, x]\).
You will frequently encounter this function when working with problems involving normal distributions or Gaussian processes. Since the erf function does not have a simple antiderivative, it is often evaluated using numerical methods or approximations.

complementary error function

The complementary error function, denoted as \text{erfc}(x), complements the error function. It is defined as: \[ \text{erfc}(x) = 1 - \text{erf}(x) \]
The complementary error function is useful because it measures the probability outside the interval \([-x, x]\).
In the original exercise, the integral is rewritten using \text{erfc}(x), simplifying the calculation. This approach often makes it easier to find the value for the integral by looking up standard tables or using numerical tools.

asymptotic series

An asymptotic series is a way of approximating a function as the argument tends toward a particular point, often infinity. It provides insight into the behavior of functions at large values.
These series are particularly useful when exact forms are hard to obtain, or when more straightforward approximations are sufficient for practical purposes.
In many cases, the asymptotic expansion can give us a quick way to estimate integral values for large limits.

power series

A power series represents a function as an infinite sum of terms, each raised to a power and multiplied by a coefficient. It is expressed as: \[ f(x) = \text{∑}_{n=0}^{∞} a_n x^n \] where \({a_n}\) are the coefficients.
Power series are incredibly useful in calculus for approximating functions, solving differential equations, and evaluating integrals.
They allow us to express complex functions in a form that is easier to work with and understand, providing a way to approximate the value of integrals when looking up exact values might be challenging.