Problem 7 - Applied Calculus

Question

Find each indefinite integral. \(\int e^{-2 y} d y\)

Step-by-Step Solution

Answer

\(-\frac{1}{2} e^{-2y} + C\)

Step 1: Identify the Integral Form

The given integral is \( \int e^{-2y} \, dy \). This integral involves an exponential function with a linear function in its exponent, which suggests a potential use of the basic integral formula for exponentials.

Step 2: Recall the Basic Exponential Integration Formula

Recall that the basic formula for integrating an exponential of the form \(\int e^{ax} \, dx\) is \(\frac{1}{a} e^{ax} + C\), where \(C\) is the constant of integration and \(a\) is a constant.

Step 3: Apply the Integration Formula

In the integral \(\int e^{-2y} \, dy\), \(a = -2\). Using the integration formula, the indefinite integral becomes \(\frac{1}{-2} e^{-2y} + C\).

Step 4: Simplify the Result

Simplify the result from Step 3 by writing it as \(-\frac{1}{2} e^{-2y} + C\). This is the antiderivative of \(e^{-2y}\) with respect to \(y\).

Key Concepts

Exponential Functions

Exponential functions are among the most important types of functions in mathematics. They have the form \(f(x) = a \cdot e^{bx}\), where \(e\) is the base of the natural logarithm, approximately equal to 2.71828, and \(a\) and \(b\) are constants. These functions grow at a rate proportional to their current value, making them ubiquitous in modeling real-world phenomena like population growth, radioactive decay, and interest calculations.
In calculus, exponential functions are special because they have the unique property that their derivative is proportional to the original function. This property also simplifies the process of integration, making them easier to manipulate in both differential and integral calculus.

Basic Integration Formula

When dealing with integration, the basic integration formula for exponentials is an essential tool. It is used to find the integral or antiderivative when given a function of the form \(e^{ax}\). The formula is:

• \(\int e^{ax} \, dx = \frac{1}{a} e^{ax} + C\)

Where \(a\) is a constant, and \(C\) is the constant of integration. This constant arises because indefinite integrals represent a family of functions, each differing by a constant.
This formula tells us that integrating an exponential function essentially reverses the differentiation process, with the additional step of dividing by the constant \(a\) from the exponent. It is a fundamental formula that simplifies the integration process significantly for exponential functions.

Antiderivatives

Antiderivatives, sometimes referred to as indefinite integrals, are functions that describe the original function whose derivative gives the function you started with. In other words, if you have a function \(f(x)\), finding its antiderivative means finding a function \(F(x)\) such that \(F'(x) = f(x)\).
For the integral \(\int e^{-2y} \, dy\), the antiderivative is \(-\frac{1}{2} e^{-2y} + C\), where \(C\) is the constant of integration. This expression represents a whole family of functions, as the constant \(C\) can be any real number.
Understanding antiderivatives is crucial because they allow us to solve problems involving areas under curves and solve differential equations, both of which are common in physics and engineering applications. They provide the reverse operation of differentiation.