Problem 52 - Calculus and its applications

Question

Evaluate. $$ \int_{0}^{b} 2 e^{-2 x} d x $$

Step-by-Step Solution

Answer

The value of the integral is \( 1 - e^{-2b} \).

Step 1: Identify the Integral Type

The given integral is a definite integral of the form \( \int_{0}^{b} 2 e^{-2x} \, dx \). This is an exponential function which can be integrated using the integration rules for exponential functions.

Step 2: Determine the Antiderivative

The function inside the integral is \( 2 e^{-2x} \). To find the antiderivative, use the formula \( \int e^{ax} \, dx = \frac{1}{a} e^{ax} + C \). For \( 2 e^{-2x} \), the antiderivative is \(-e^{-2x} + C\).

Step 3: Apply Definite Integral Limits

Now apply the limits of integration from 0 to \( b \) using the Fundamental Theorem of Calculus. Substitute the upper limit \( b \) and the lower limit 0 into the antiderivative. This results in: \(-e^{-2b} - (-e^{0})\).

Step 4: Simplify the Expression

Calculate the terms: \(-e^{-2b} - (-e^{0})\) simplifies to \(-e^{-2b} + 1\) because \( e^{0} = 1 \). Therefore, the result of the integral is: \( 1 - e^{-2b} \).

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where the variable is located in the exponent. This makes them grow or decay quite rapidly as the variable changes. They're used in scenarios where things increase or decrease at consistent rates, such as population growth or radioactive decay.
In equations, exponential functions commonly appear as \(e^{ax}\), where \(a\) is a constant. For example, in the function \(2e^{-2x}\), the exponential term is \(e^{-2x}\). Here, the base \(e\), approximately equal to 2.718, is raised to the power of \(-2x\).

• Exponential functions are crucial in calculus for modeling real-world phenomena.
• They have distinct characteristics like always being positive and having a slope that is proportional to the function itself.

If you encounter exponential expressions in integrals, special integration rules apply for solving them, making understanding these functions essential.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a bridge between differential calculus and integral calculus. It connects the concept of a derivative of a function with the concept of an integral. Essentially, it tells us that \( ext{if} \, F(x) \, ext{is an antiderivative of} \, f(x), \, ext{then} \, \int_a^b f(x) \, dx = F(b) - F(a).\).
With this theorem, you can evaluate definite integrals quickly once you know the antiderivative.

• The theorem simplifies the process by eliminating the need to calculate the area under a curve from scratch.
• It highlights the inverse relationship between differentiation and integration.

In the exercise, this theorem allows us to substitute the upper and lower limits into the antiderivative, offering a direct result. This saves time and effort, giving us efficient solutions.

Antiderivatives

Antiderivatives are the opposite of derivatives. While a derivative indicates the rate of change, an antiderivative reverses this process. Simply put, if you know the derivative of a function, you can find the original function (possibly plus a constant) that would yield that derivative.
For example, finding the antiderivative of \(2e^{-2x}\) involves reversing the differentiation process which results in \(-e^{-2x} + C\). The \(+ C\) represents any constant which wouldn't affect the derivative.

• Antiderivatives are crucial for finding definite integrals by applying the Fundamental Theorem of Calculus.
• Recognizing common patterns in derivatives helps you find antiderivatives quicker.

Mastering antiderivatives is key to solving many integral-related problems effectively.