Problem 40 - Applied Calculus

Question

(a) Evaluate \(\int_{0}^{b} x e^{-x / 10} d x\) for \(b=10,50,100,200\). (b) Assuming that it converges, estimate the value of \(\int_{0}^{\infty} x e^{-x / 10} d x\).

Step-by-Step Solution

Answer

For large b, the integral converges to 0.

Step 1: Identify the Integration Type

The given integral \( \int_{0}^{b} x e^{-x / 10} \, dx \) is a definite integral involving the exponential function multiplied by \( x \). This is a case for integration by parts.

Step 2: Integration by Parts Formula

The formula for integration by parts is \( \int u \, dv = uv - \int v \, du \). We need to choose \( u = x \) and \( dv = e^{-x/10} \, dx \).

Step 3: Differentiate and Integrate

Differentiate \( u = x \) to get \( du = dx \). Integrate \( dv = e^{-x/10} \, dx \) to get \( v = -10e^{-x/10} \).

Step 4: Apply the Integration by Parts Formula

Substitute these into the integration by parts formula to get: \[ \int x e^{-x/10} \, dx = -10xe^{-x/10} + 10 \int e^{-x/10} \, dx. \]

Step 5: Solve the Remaining Integral

The remaining integral is \( \int e^{-x/10} \, dx = -10e^{-x/10} \). So, the expression becomes: \[ -10xe^{-x/10} - 100e^{-x/10} + C. \]

Step 6: Evaluate the Definite Integral from 0 to b

Now evaluate the expression at \( x = 0 \) and \( x = b \). Substituting these limits in, we get: \[ (-10be^{-b/10} - 100e^{-b/10}) - (0) = -110e^{-b/10} (b + 10). \]

Step 7: Calculate for Given Values of b

Evaluate for \( b = 10, 50, 100, 200 \): - For \( b = 10 \), the result is \(-110e^{-1}(20)\).- For \( b = 50 \), the result is \(-110e^{-5}(60)\).- For \( b = 100 \), the result is \(-110e^{-10}(110)\).- For \( b = 200 \), the result is \(-110e^{-20}(210)\).

Step 8: Evaluate Convergence for b Approaching Infinity

Establish convergence by taking \( b \to \infty \). The term \( e^{-b/10} \to 0 \) as \( b \to \infty \), hence the integral converges to \( 0 \).

Key Concepts

Definite Integral

A definite integral is a way of calculating the area under a curve, from one point to another. It represents the accumulated quantity between these two points. In our exercise, the definite integral is given as \( \int_{0}^{b} x e^{-x / 10} \, dx \), where \( 0 \) and \( b \) are the limits of integration. This integral calculates the area under the curve of the function \( x e^{-x / 10} \) from \( x = 0 \) to \( x = b \).

Unlike indefinite integrals, which have a family of solutions with a constant \( C \), definite integrals provide a specific value that depends on the interval \([a, b]\). In this problem, the result depends on the choice of \( b \), and as \( b \) changes, the area, or the scalar value of the integral, changes accordingly. Calculations were performed for different values of \( b \), showcasing how the definite integral's outcome changes as \( b \) increases. For example:

• For \( b = 10 \), the definite integral produced a specific result.
• Similarly, the result changes as \( b \) changes to \( 50, 100, \) and \( 200 \).

This understanding is critical for solving problems involving definite integrals, as it gives insight into how integrals work to measure areas and accumulated quantities in real-life situations.

Exponential Function

An exponential function is one where a constant base is raised to a variable exponent. In our exercise, the exponential function is \( e^{-x/10} \), where \( e \) is the usual Euler's number, approximately 2.718. The function \( e^{-x/10} \) shows how the value changes rapidly with changes in \( x \).

Exponential functions, especially those involving time decay or growth models, are widely used in fields like physics, finance, and engineering. In this case, \( e^{-x/10} \) is part of our integral equation, creating a decaying effect on \( x \). It decreases exponentially as \( x \) increases, which means the larger the value of \( x \), the smaller \( e^{-x/10} \) gets.

• This property is useful because it highlights the role of exponential decay in the behavior of the definite integral.
• As seen in the solutions, at smaller values of \( b \), there are noticeable results, while at larger \( b \), the effect diminishes due to this rapid decay.

Recognizing and manipulating exponential functions is essential for solving higher-level mathematics problems, as they frequently appear in calculus and differential equations.

Convergence of Integrals

Convergence is a key concept in calculus, especially when dealing with integrals. An integral converges if its value approaches a finite number as the upper limit of integration goes to infinity. In our exercise, we examine the limit as \( b \to \infty \).

The expression \( -110e^{-b/10}(b + 10) \) shows us how the integral behaves as \( b \) increases. Since \( e^{-b/10} \to 0 \) when \( b \to \infty \), the entire expression approaches zero, leading to the conclusion that the integral converges. This convergence means that the accumulated area under the curve \( x e^{-x/10} \) from \( x = 0 \) to \( x = \infty \) settles to a finite value, improving our understanding of the function's behavior over an infinite interval.

• Convergent integrals are critical in mathematics and applied sciences because they imply stability and a bounded solution.
• Knowing when and how integrals converge influences fields like probability, where stability and expected values are important.

In conclusion, assessing convergence helps evaluate whether the behavior of a function over a prolonged interval leads to meaningful results.