Question

When integrating \(f_{x}(x, y)\) with respect to \(x\), the constant of integration \(C\) is really which: \(C(x)\) or \(C(y) ?\) What does this mean?

Short answer

The constant of integration is \(C(y)\), indicating it can depend on \(y\).

Step-by-step solution

Understand the Function to Integrate

We are given a function \(f(x, y)\), which is a function of two variables: \(x\) and \(y\). Our task is to integrate this function with respect to \(x\). This means \(y\) is treated as a constant during the integration.

Setting Up the Integration

When integrating \(f(x, y)\) with respect to \(x\), we treat \(y\) as a constant. This is similar to integrating a single-variable function, with the exception that \(y\) is simply a constant term in the integration process.

Determining the Constant of Integration

Upon integrating with respect to \(x\), the constant of integration, commonly denoted as \(C\), arises from the indefinite integration. Since the integration is performed with respect to \(x\), any terms dependent solely on \(y\) will not change when differentiating the result with respect to \(x\). Hence, the constant of integration \(C\) can be a function of \(y\), written as \(C(y)\).

Interpretation of the Result

The constant of integration \(C(y)\) implies that when you differentiate the integrated function with respect to \(x\), \(C(y)\) disappears since \(y\) is constant concerning \(x\). Therefore, \(C\) could depend on \(y\), adapting to any functional relationship \(y\) might have within the context of the problem.

Key concepts

Multivariable Calculus

In mathematics, multivariable calculus extends the concepts of calculus into multiple dimensions. In contrast to single-variable calculus, which deals with functions of one variable, multivariable calculus allows us to work with functions that depend on several variables. For example, a function like \( f(x, y) \) depends on two variables, \( x \) and \( y \).

When integrating a function of multiple variables, the process can differ slightly compared to single-variable functions. If you are integrating with respect to one variable, say \( x \), the other variable \( y \) is treated as a constant. This designation is key in understanding how multivariable calculus allows us to take derivatives or integrals, even when the function's behavior is influenced by other dimensions.

Multivariable calculus is vital in fields like physics and engineering, where systems depend on various parameters. Mastering these concepts can give you an edge when tackling real-world problems involving complex systems.

Constant of Integration

In calculus, the constant of integration is a crucial component of indefinite integrals. When integrating, especially with indefinite integrals, we often add a constant term, represented by \( C \). This constant accounts for the family of functions that differ by a constant but have the same derivative.

In multivariable calculus, identifying \( C \) is crucial, especially when dealing with functions with more than one variable. When integrating a function like \( f(x, y) \) with respect to \( x \), this constant of integration can be a function of the other variable, in this case, \( y \), thus represented as \( C(y) \). This is because \( y \) remains constant during the integration process.

Understanding why \( C \) can depend on other variables helps maintain the functional relationship within the context of a multivariable system and ensures accurate integration results.

Indefinite Integral

An indefinite integral represents a family of functions and is the reverse process of taking a derivative. Unlike definite integrals, which compute a net area under a curve, indefinite integrals lack limits of integration and instead describe a general form of antiderivatives that differs by a constant.

Such integrals are represented as \( \int f(x) \ dx \), leading to a result where a constant of integration, \( C \), is added. In the realm of multivariable calculus, when integrating with respect to one variable (like \( x \) in \( f(x, y) \)), the constant \( C \) becomes a function of the other variable, \( y \).

Indefinite integrals are essential in solving differential equations and restoring original functions from their rates of change. Recognizing their broader implications in multivariable settings equips students with a fundamental understanding of mathematical modeling in complex scenarios.