Problem 23 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(1-34\) without using tables. $$ \int_{-\infty}^{0} e^{-|x|} d x $$

Step-by-Step Solution

Answer

The integral evaluates to 1.

Step 1: Recognize the Nature of the Function

The integral involves the absolute value function inside the exponential. For \(x < 0\), \(|x| = -x\), so this can be simplified as \(e^{-|x|} = e^{x}\) for this interval.

Step 2: Set Up the Integral

Since the integral is from \(-\infty\) to 0, we rewrite the integral using our understanding from Step 1:\[\int_{-finity}^{0} e^{-(-x)} \, dx = \int_{-finity}^{0} e^{x} \, dx.\]

Step 3: Integrate the Function

The integral of \(e^{x}\) with respect to \(x\) is itself \(e^{x}\). Therefore,\[\int e^{x} \, dx = e^{x} + C,\]where \(C\) is the constant of integration, but will not affect this as we evaluate a definite integral.

Step 4: Evaluate the Definite Integral

Evaluate the integral from \(-\infty\) to 0:\[\left[ e^{x} \right]_{-finity}^{0} = e^{0} - \lim_{x \to -\infty} e^{x}.\]Since \(e^{0} = 1\) and \(\lim_{x \to -\infty} e^{x} = 0\), the integral evaluates to \(1 - 0 = 1\).

Step 5: Conclude the Result

The integral \(\int_{-finity}^{0} e^{-|x|} \, dx\) evaluates to 1, as shown by the calculations in the preceding steps.

Key Concepts

Definite Integrals

Definite integrals are a way to find the area under a curve over a specific interval. In this case, we are looking at the interval from \(-\infty\) to 0. Unlike indefinite integrals, which include a constant of integration, definite integrals evaluate to a fixed number.
This is because the limits of integration, the numbers at the top and bottom of the integral sign, provide a boundary. You can think of it as finding the total accumulation of quantities, such as distance, area, or population, between two points.

• The lower limit is \(-\infty\), suggesting we are calculating from far left down the x-axis.
• The upper limit is 0, showing where we stop the calculation.

The solution doesn't need a constant because definite integrals represent a specific value: the entire accumulated quantity across the interval. For the integral \(\int_{-\infty}^{0} e^{x} \, dx\), it helped us find a confirmed numeric answer of 1.

Exponential Functions

Exponential functions are expressions where the variable, typically \(x\), is an exponent. A common base for these exponents is \(e\), Euler's number, which is approximately 2.718.
The formula for the exponential function is \(e^{x}\), which increases rapidly as \(x\) becomes larger.
Notably, the derivative and integral of \(e^{x}\) are both \(e^{x}\), making it unique among functions and quite practical for integration and differentiation.

• This property simplifies calculations, often leading directly to the result of the definite integral.
• In the example \( \int_{-\infty}^{0} e^{x} \, dx \), this direct correlation helps find that the integral is 1.

Understanding this nature of exponential functions allows for seamless calculus operations, crucial when evaluating such integrals.

Absolute Value Integration

The absolute value function, denoted \(|x|\), affects the computation of integrals by its nature to return non-negative values. In our example, the presence of \(e^{-|x|}\) converts based on the range of \(x\). This changes negatively expressed components into positive ones.
Here's how it operates:

• Typically, \(|x|\) equals \(-x\) when \(x < 0\), and this becomes particularly relevant from integration over negative intervals, like in this problem from \(-\infty\) to 0.
• When evaluating \(e^{-|x|}\) for \(x < 0\), it simplifies to \(e^{x}\) as observed in the original solution.

Absolute values can initially appear complex within integrals, but carefully recognizing the nature of \(x\) enables smooth transitions, streamlining the process of evaluating such integrals.