Problem 30 - Calculus: Early Transcendental Functions

Question

Use the Integral Mean Value Theorem to estimate the value of the integral. $$\int_{0}^{1 / 2} e^{-x^{2}} d x$$

Step-by-Step Solution

Answer

The estimated value of the integral from 0 to 1/2 of \(e^{-x^{2}}\) dx is 1/2.

Step 1: Identify the function and interval

From the problem, we can identify that the function is \(f(x) = e^{-x^{2}}\) and the interval is [0, 1/2]

Step 2: Find the maximum value of the function on the interval

The maximum value of the function \(e^{-x^{2}}\) on the interval [0, 1/2] occurs at x = 0. Hence, the maximum value is f(0) = \(e^{-0}\) = 1

Step 3: Apply the Integral Mean Value Theorem

The Integral Mean Value Theorem tells us that the integral from a to b of f(x) dx equals to f(c) times (b - a) where c is the value in the interval [a, b] at which f reachers its maximum value. We already know from step 2 that the maximum value is at c = 0. Hence, the integral is f(c) * (b - a) = 1 * (1/2 - 0) = 1 * 1/2 = 1/2.

Key Concepts

Definite Integral

A definite integral is a concept in calculus that represents the accumulation of quantities, such as areas under a curve, over a specified interval. In our exercise, the integral \( \int_{0}^{1 / 2} e^{-x^{2}} \, dx \) signifies the area under the curve of the function \( e^{-x^{2}} \) from \( x=0 \) to \( x=1/2 \).

The process of finding a definite integral involves evaluating the antiderivative, or the inverse operation of taking the derivative. However, not all functions have simple antiderivatives. For complex functions like \( e^{-x^{2}} \), numerical methods or approximation techniques, such as the Integral Mean Value Theorem, are often used to estimate their values.

A key aspect of definite integrals is that they provide a precise formulation of the net accumulation over an interval, rather than an indefinite integral which lacks boundary limits.

Exponential Function

Exponential functions are a class of functions where the variable appears in the exponent, such as \( f(x) = e^{x} \).

In our problem, the specific exponential function is \( e^{-x^{2}} \). This type of function is notably useful in modeling situations involving rapid growth or decay. It exhibits unique properties like being its own derivative, mirroring its behavior under calculus operations.

The function \( e^{-x^{2}} \) is a special type of exponential called a Gaussian function, which appears frequently in statistics and probability, particularly in the context of normal distributions. Its distinct bell-shaped curve makes it useful for representing data distributions and other natural phenomena.

Function Maximum

Finding the maximum value of a function over an interval is a critical step in applying the Integral Mean Value Theorem. For a function \( f(x) \) on a closed interval \([a, b]\), the maximum value occurs at a point where either the derivative is zero or at one of the endpoints of the interval.

In our exercise, the function \( e^{-x^{2}} \) reaches its maximum value on the interval \([0, 1/2]\) at \( x = 0 \).

• This is because \( f(x) = e^{-x^{2}} \) is decreasing on this interval.
• The maximum value at \( x = 0 \) is \( e^{0} = 1 \).

Recognizing where a function achieves its maximum is essential not only for applying the Integral Mean Value Theorem, but also for other aspects of calculus, optimization, and real-world applications.