Chapter 12: Problem 2
In Exercises \(1-10\), evaluate the integral. $$ \int_{x}^{x^{2}} \frac{y}{x} d y $$
Short Answer
Expert verified
The value of the definite integral \( \int_{x}^{x^{2}} \frac{y}{x} dy \) is \( \frac{x^3}{2} - \frac{x}{2} \)
Step by step solution
01
Find the Indefinite Integral
First, perform the indefinite integral of \( \frac{y}{x} \) with respect to \( y \). The \( x \) is taken as a constant while integrating with respect to \( y \), that's why it's set outside the integral. The result is: \( \int \frac{y}{x} dy = \frac{1}{x} \int y dy = \frac{y^2}{2x} + C\) where \(C\) is the integration constant.
02
Apply the Definite Integral Evaluation Theorem
Replace the boundaries \(x\) and \(x^2\) into the equation \( \frac{y^2}{2x} \). So, the definite integral is:\( \left[\frac{y^2}{2x}\right]_x^{x^2} = \frac{(x^2)^2}{2x} - \frac{x^2}{2x}\) = \( \frac{x^4}{2x} - \frac{x^2}{2x} \).
03
Simplify the Expression
Simplify the expression further: \( \frac{x^4}{2x} - \frac{x^2}{2x} = \frac{x^3}{2} - \frac{x}{2} \). This is the value of the definite integral.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Indefinite Integral
The indefinite integral, also known as an antiderivative, is a fundamental concept in calculus. It represents the opposite process of differentiation, meaning that the integral of a function is another function whose derivative is the original function. When we integrate, we are essentially finding all possible antiderivatives of a function.
For example, if you are given a function to integrate, such as \( \frac{y}{x} \), where \( x \) is considered constant and \( y \) is the variable, you would multiply \( y \) by the reciprocal of \( x \) and integrate \( y \) with respect to \( y \) itself. This process leads to \( \frac{y^2}{2x} \), plus a constant of integration, often denoted as \( C \). This constant represents an infinite number of possible constant values that, when added to \( \frac{y^2}{2x} \), give a family of antiderivatives.
Understanding the indefinite integral is crucial because it sets the foundation for evaluating definite integrals, which provide the area under a curve within specific limits.
For example, if you are given a function to integrate, such as \( \frac{y}{x} \), where \( x \) is considered constant and \( y \) is the variable, you would multiply \( y \) by the reciprocal of \( x \) and integrate \( y \) with respect to \( y \) itself. This process leads to \( \frac{y^2}{2x} \), plus a constant of integration, often denoted as \( C \). This constant represents an infinite number of possible constant values that, when added to \( \frac{y^2}{2x} \), give a family of antiderivatives.
Understanding the indefinite integral is crucial because it sets the foundation for evaluating definite integrals, which provide the area under a curve within specific limits.
Integration
Integration is a core operation in calculus, primarily used for finding the areas under curves, the accumulation of quantities, and in many other contexts. It can be visualized as adding up an infinite number of infinitesimally small quantities.
Two primary forms of integration exist: definite and indefinite. While we've discussed the indefinite integral in the previous section, the definite integral is more about computation within exact boundaries. It is represented with lower and upper limits on the integral sign, like in \( \[ \frac{y}{x} \]_{x}^{x^2} \). This notation indicates that we're interested in finding the total accumulation of \( \frac{y}{x} \) as \( y \) changes from \( x \) to \( x^2 \). The process of integrating can be achieved through various methods, including basic integration rules, integration by parts, and integration by substitution.
Two primary forms of integration exist: definite and indefinite. While we've discussed the indefinite integral in the previous section, the definite integral is more about computation within exact boundaries. It is represented with lower and upper limits on the integral sign, like in \( \[ \frac{y}{x} \]_{x}^{x^2} \). This notation indicates that we're interested in finding the total accumulation of \( \frac{y}{x} \) as \( y \) changes from \( x \) to \( x^2 \). The process of integrating can be achieved through various methods, including basic integration rules, integration by parts, and integration by substitution.
Calculus
Calculus is an advanced field of mathematics that deals with change and motion. Divided into two main branches, differential calculus and integral calculus, it provides a framework for modeling and examining dynamic systems.
Differential calculus focuses on the rate of change (derivatives), whereas integral calculus concerns the accumulation of quantities (integrals). The connection between these two, known as the Fundamental Theorem of Calculus, states that differentiation and integration are inverse processes. Mastery of calculus is essential in various scientific disciplines, including physics, engineering, economics, and more, as it allows the formulation and solving of complex problems involving change. In the context of integration, applying calculus to solve problems often involves evaluating integrals to find areas, distances, volumes, and other physical quantities.
Differential calculus focuses on the rate of change (derivatives), whereas integral calculus concerns the accumulation of quantities (integrals). The connection between these two, known as the Fundamental Theorem of Calculus, states that differentiation and integration are inverse processes. Mastery of calculus is essential in various scientific disciplines, including physics, engineering, economics, and more, as it allows the formulation and solving of complex problems involving change. In the context of integration, applying calculus to solve problems often involves evaluating integrals to find areas, distances, volumes, and other physical quantities.
Integration by Substitution
Integration by substitution is a powerful technique for evaluating integrals that may not be immediately solvable by standard methods. It involves changing the variable of integration to simplify the integral into a more familiar form, which can then be integrated with respect to the new variable.
This method is analogous to applying the chain rule of differentiation in reverse. For instance, when the integrand includes a function and its derivative, we can use substitution to make the integral straightforward. The process involves selecting a substitution that makes the integral simpler, replacing the variables, and then integrating with respect to the new variable.
In the given exercise \( \int_{x}^{x^{2}} \frac{y}{x} d y \) the substitution method wasn't necessary as the integral is already in a simple form. However, it's a tool that's incredibly useful for more complex integrals where direct integration isn't feasible.
This method is analogous to applying the chain rule of differentiation in reverse. For instance, when the integrand includes a function and its derivative, we can use substitution to make the integral straightforward. The process involves selecting a substitution that makes the integral simpler, replacing the variables, and then integrating with respect to the new variable.
In the given exercise \( \int_{x}^{x^{2}} \frac{y}{x} d y \) the substitution method wasn't necessary as the integral is already in a simple form. However, it's a tool that's incredibly useful for more complex integrals where direct integration isn't feasible.
