Problem 52 - Calculus for Biology and Medicine

Question

$$ \text { Find } \int_{-3}^{-3} e^{-x^{2} / 2} d x \text { . } $$

Step-by-Step Solution

Answer

The integral is 0 because the limits of integration are the same.

Step 1: Understanding the Problem

We are given the integral \( \int_{-3}^{-3} e^{-x^{2} / 2} \, dx \). The task is to evaluate this definite integral.

Step 2: Identifying the Special Case

Notice that the limits of integration are both \(-3\). In general, for any function \( f(x) \), the definite integral \( \int_a^a f(x) \, dx = 0 \) because the interval length is zero.

Step 3: Applying the Rule

Applying this rule, we have \( \int_{-3}^{-3} e^{-x^{2} / 2} \, dx = 0 \).

Key Concepts

Definite Integral

A definite integral is an essential tool in calculus used to calculate the accumulation of values, or the area under a curve from a specific starting point to an ending point. In mathematical terms, it is written as \( \int_a^b f(x) \, dx \). Here, \( a \) and \( b \) represent the limits of integration, and \( f(x) \) is the function being integrated.

Definite integrals can give different types of results based on the function's behavior over the interval:

• The area under the curve of \( f(x) \) between \( a \) and \( b \).
• Negative areas if the function is below the x-axis over the interval.

They can be utilized in a variety of real-world applications such as calculating distances, areas, volumes, and even probabilities. Understanding how to compute definite integrals helps to connect the concept of integration with practical scenarios, enhancing the ability to solve complex problems.

Limits of Integration

The limits of integration, \( a \) and \( b \), define the interval over which the definite integral is computed. They specify the start and end points along the x-axis for evaluating the area under the curve or sum of values.

An interesting property of limits of integration is when they are equal, meaning \( a = b \).
This creates a situation where the length of the interval is zero because there is no distance between the start and end point. Hence, the definite integral \( \int_a^a f(x) \, dx \) will always result in zero, regardless of the function \( f(x) \), because there is no area to calculate.

Manipulating these limits can greatly influence the result of an integral. Practicing with different limits helps grasp this concept fully and understanding how changing limits impacts the results is crucial in solving integration problems accurately.

Special Cases in Integration

In the world of integration, there are special cases that simplify problems and provide quick solutions when certain conditions are met. One such special case occurs when the limits of integration are identical.

As noted before, if the limits are the same, say both are \(-3\), then the definite integral yields zero. This is due to the fact there is essentially no interval to integrate over. Recognizing these special cases can save time and effort when evaluating integrals.

Another interesting aspect of special cases might involve symmetrical functions over the limits that simplify the evaluation. If a problem involves symmetry between the limits, sometimes the integral can be broken down into simpler parts.

Understanding these nuances provides shortcuts and deeper insights into what integrals represent and how they can be evaluated efficiently. Being aware of these can significantly enhance your problem-solving skills, particularly in more elaborate calculus problems.