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Chapter 7: Problem 2

Evaluate the partial integral. $$ \int_{x}^{x^{2}} \frac{y}{x} d y $$

Short Answer

Expert verified
The result of the integral \(\int_{x}^{x^{2}} \frac{y}{x} dy \) is \( \frac{x^3 - x}{2} \).

Step by step solution

01

Identify the Function to be Integrated

Our function to be integrated here is \( \frac{y}{x} \). The variable of integration is \( y \), meaning we are integrating with respect to \( y \). Our limits of integration are from \( x \) to \( x^2 \).
02

Compute the Definite Integral

The integral is computed as follows: \[ \int_{x}^{x^{2}} \frac{y}{x} dy \]. Notice here \( x \) is a constant because we are integrating with respect to \( y \). We can simplify the integral by pulling out the \( \frac{1}{x} \) since it is a constant relative to \( y \). This gives us: \[ \frac{1}{x} \int_{x}^{x^{2}} y dy \]. Now, evaluate this integral. The integral of \( y \) with respect to \( y \) is \( \frac{1}{2} y^2 \). We have then: \[ \frac{1}{2x} [ (x^2)^2 - x^2 ] \]
03

Simplify and Solve

Simplify the above expression. \( (x^2)^2 = x^4 \), so the expression becomes \[ \frac{1}{2x} [ x^4 - x^2 ] \]. This simplifies to \( \frac{x^3 - x}{2} \). Thus, the solution to the integral is \( \frac{x^3 - x}{2} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Definite Integral
A definite integral is a way to calculate the area under a curve within specified limits. It's written as \( \int_{a}^{b} f(x) \, dx \), where \( a \) and \( b \) are the limits of integration and \( f(x) \) is the function being integrated.
The term 'definite' distinguishes it from an indefinite integral, which lacks specific limits of integration and usually includes an arbitrary constant.
In the process of integrating, we find a function (the antiderivative) whose derivative is the integrand, and then evaluate this antiderivative at the upper and lower limits:
  • First, find the antiderivative of the function.
  • Then, evaluate it at the upper and lower limits of integration.
  • Subtract the two results to get the area under the curve between these two limits.
Limits of Integration
The limits of integration are the two values that define the starting and ending points of an integral. They are crucial in determining the "piece" of the function you are interested in.
In our problem, the limits are \( x \) and \( x^2 \). This means we are interested in evaluating the integral from when \( y = x \) to when \( y = x^2 \).
These limits specify the interval over which the integration is performed. Limits can be constants, but in this case, they are expressed in terms of another variable, \( x \). Understanding them helps with:
  • Identifying where the integration begins and ends.
  • Determining what part of the graph or curve of the function is being considered.
  • Ensuring proper evaluation of the integral by plugging these limits into the antiderivative.
Variable of Integration
The variable of integration is the variable with respect to which the function is being integrated. It determines which variable's change is being considered in the integration process.
In our example, the variable of integration is \( y \). This means we treat all other variables, such as \( x \), as constants during the integration process.
Understanding the variable of integration helps in:
  • Identifying which variable is active, or varying, in the integral.
  • Treating other variables as constants, simplifying complex expressions.
  • Applying the correct integration techniques based on which variable is chosen.
By carefully tracking the variable of integration, one can effectively compute more complex integrals, ensuring accuracy and simplifying calculations.

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