Problem 1 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-24, evaluate each improper integral or show that it diverges. \(\int_{100}^{\infty} e^{x} d x\)

Step-by-Step Solution

Answer

The integral \( \int_{100}^{\infty} e^{x} \, dx \) diverges.

Step 1: Understand the Nature of the Integral

The given integral is improper because it has an upper limit of infinity. We'll need to evaluate this improper integral or show that it does not converge.

Step 2: Set Up the Limit Definition

Since the upper bound is infinity, we express the integral as a limit:\[\int_{100}^{ ext{b}} e^x \, dx\] and then take the limit as \( b \to \infty \).

Step 3: Evaluate the Indefinite Integral

The antiderivative of \( e^x \) is \( e^x \). Thus, the integral becomes:\[\int e^x \, dx = e^x + C\]where \( C \) is the constant of integration.

Step 4: Apply the Limits

Substitute the limits from 100 to \( b \) into the antiderivative:\[[e^x]_ {100}^{b} = e^b - e^{100}\]

Step 5: Evaluate the Limit as b Approaches Infinity

Now take the limit as \( b \to \infty \):\[\lim_{{b \to \infty}} (e^b - e^{100})\]As \( b \to \infty \), \( e^b \to \infty \). Therefore, the limit is infinite indicating divergence.

Key Concepts

Divergence of Integrals

When evaluating integrals, especially improper ones, you'll need to determine if they diverge or converge. Divergence occurs when the integral does not settle to a finite value.
In the context of improper integrals with an infinite limit, this means that the evaluation approaches infinity rather than stabilizing.

• For example, in this exercise, the function being integrated, \( e^x \), grows exponentially.
• This is faster than it can be brought to a finite sum as \( x \) approaches infinity, leading to divergence.

Recognizing divergence is vital, as it tells you that the area under the curve grows indefinitely.
This is a key aspect of analyzing improper integrals.

Limit of a Function

Limits are essential in understanding behavior near a point, especially as variables approach infinity.
When dealing with improper integrals, converting the integral into a limit expression helps determine convergence or divergence.

• For our problem, the function \( e^x \) is evaluated from 100 to \( b \), with a limit taken as \( b \to \infty \).
• This technique is crucial because the behavior at \( b \to \infty \) informs us about the existence of a finite area under the curve.

Such a process allows us to convert an infinite scenario into an evaluative process using limits.
Limits simplify analysis by providing a mechanism to model and understand infinite behaviors.

Indefinite Integral

Before applying limits, we first need to evaluate the indefinite integral, or the antiderivative, of the given function.
The indefinite integral gives us the general form of the function whose derivative is the integrand.

• In this problem, the antiderivative of \( e^x \) is simply \( e^x + C \), where \( C \) represents the constant of integration.
• This step is critical in forming the basis of evaluating the definite integral.

It helps visualize the integral's outcome over a defined range, allowing for limits to be applied.
Understanding the indefinite integral is foundational before tackling improper integrals.

Antiderivatives

Antiderivatives are reversed operations of derivatives, fundamental to integral calculus.
They provide the "opposite" of differentiation and are integral (pun intended) to solving integrals.

• For the function \( e^x \), recognizing that its antiderivative is \( e^x \) simplifies solving the integral bounds.
• This knowledge streamlines the substitution phase in our definite integral evaluation.

Mastering antiderivatives is essential for successfully tackling integrals, both definite and indefinite.
They form the backbone upon which more complex calculus operations, including solving improper integrals, are built.