Problem 54 - Calculus and its applications

Question

Evaluate. $$ \int_{0}^{b} k e^{-k x} d x $$

Step-by-Step Solution

Answer

The integral evaluates to \( 1 - e^{-kb} \).

Step 1: Understand the Integral

We are given the integral \( \int_{0}^{b} k e^{-kx} \, dx \), where \( k \) is a constant and \( b \) is the upper limit of integration. This is an exponential integral that typically involves substitution for simplification.

Step 2: Choose a Substitution

Let \( u = -kx \). Then, differentiate to find \( du = -k \, dx \), or equivalently, \( dx = -\frac{1}{k} \, du \). Substitute these into the integral.

Step 3: Change the Limits of Integration

When \( x = 0 \), \( u = -k \times 0 = 0 \). When \( x = b \), \( u = -kb \). Thus, the new limits of integration are from \( 0 \) to \( -kb \).

Step 4: Substitute and Simplify the Integral

Substitute \( u = -kx \) and \( dx = -\frac{1}{k} \, du \) into the integral:\[\int_{0}^{-kb} k e^{-kx} \, dx = \int_{0}^{-kb} e^{u} \left(-\frac{1}{k}\right) \, du = -\int_{0}^{-kb} e^{u} \, du.\]

Step 5: Evaluate the Integral

The integral \(-\int_{0}^{-kb} e^{u} \, du\) can be evaluated as:\[= -[e^{u}]_{0}^{-kb} = -(e^{-kb} - e^{0}) = -(e^{-kb} - 1).\]

Step 6: Simplify and Finalize

The final expression simplifies to:\[1 - e^{-kb}.\]Thus, the evaluated integral is \( 1 - e^{-kb} \).

Key Concepts

Exponential Integral

An exponential integral involves the integration of an expression containing an exponential function, typically of the form \( e^{ax} \) where \( a \) is a constant. These types of integrals appear frequently in calculus, especially when dealing with natural processes or decay problems that can be modeled by exponential functions. Understanding exponential integrals is essential for solving problems that involve continuous growth or decay. In our exercise, we had the function \( k e^{-kx} \), where \( k \) is a constant and \( e^{-kx} \) represents an exponential decay:

• The presence of the negative sign in the exponent indicates that the function decays as \( x \) increases.
• These integrals often require specific techniques, such as integration by substitution, to simplify the computation.

Exponential integrals are widely applicable in real-life scenarios like calculating the cooling of an object or the discharging of a capacitor. They provide deep insights into how exponential functions behave under integration, leading to results that describe the accumulated effect over an interval.

Integration by Substitution

Integration by substitution is a fundamental technique used to simplify complex integrals. It is analogous to the reverse process of the chain rule in differentiation. The main idea is to substitute a part of the integral with a new variable, making the integral easier to evaluate. Here's how it works:

• Identify a component of the integral that, when replaced by a new variable, simplifies the integrand.
• In our exercise, we chose \( u = -kx \). This substitution helps because it simplifies the exponential function into \( e^{u} \).
• Once the substitution is made, differentiate to find \( du \), and express \( dx \) in terms of \( du \).
• Don't forget to change the limits of integration to match the new variable, as we did when converting \( x \) limits to \( u \) limits: from \( 0 \) to \(-kb\).

This method can transform a difficult problem into a straightforward one by making the integrand something familiar, typically allowing for a direct evaluation or a simple antiderivative.

Calculus Problem Solving

Calculus problem solving involves applying various techniques and methods to evaluate integrals, differentiate functions, or solve mathematical models. When faced with an integral like \( \int_{0}^{b} k e^{-kx} \, dx \), a structured approach is essential for successful problem solving:

• Start by analyzing the integral to understand its components and potential difficulties.
• Determine the most suitable integration technique. In this case, integration by substitution was used to handle the exponential function effectively.
• Change variables if needed, adjust integration limits appropriately, and carry out the substitution.
• After substitution, simplify the integral to find a more tractable form, such as \( \int_{0}^{-kb} e^{u} \, du \).
• Finally, calculate the antiderivative and return to the original variables if required, ensuring the solution is expressed in terms of the given limits and terms.

This meticulous step-by-step process ensures that you capture all transformations, leading to correct and logical solutions. It highlights the importance of understanding each component of the problem and executing the appropriate strategy for evaluation.