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Chapter 7: Problem 9

Evaluate each iterated integral. $$ \int_{0}^{2} \int_{0}^{1} x d y d x $$

Short Answer

Expert verified
The evaluated integral is 2.

Step by step solution

01

Analyze the Integral

We are given the double integral \( \int_{0}^{2} \int_{0}^{1} x \, d y \, d x \). We'll integrate the function \( x \) first with respect to \( y \) and then with respect to \( x \).
02

Integrate with Respect to \( y \)

To evaluate the inner integral \( \int_{0}^{1} x \, d y \), we treat \( x \) as a constant because it does not depend on \( y \). The integral becomes \( x \int_{0}^{1} \, d y = x [y]_{0}^{1} = x(1 - 0) = x \).
03

Simplify the Integral

After integrating with respect to \( y \), the double integral simplifies to \( \int_{0}^{2} x \, d x \) as \( x \) is the result of the inner integral.
04

Integrate with Respect to \( x \)

Now, evaluate the outer integral \( \int_{0}^{2} x \, d x \). This requires finding the antiderivative of \( x \): \( \frac{x^2}{2} \). We will evaluate it using the limits from 0 to 2.
05

Evaluate the Antiderivative

Plug the limits into \( \frac{x^2}{2} \): \[ \left. \frac{x^2}{2} \right|_{0}^{2} = \frac{2^2}{2} - \frac{0^2}{2} = \frac{4}{2} - 0 = 2 \].
06

Conclusion

The value of the original double integral \( \int_{0}^{2} \int_{0}^{1} x \, d y \, d x \) is 2.

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Key Concepts

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

Iterated Integrals
Iterated integrals let us evaluate functions over two-dimensional regions. In this scenario, we first integrate with respect to one variable, then the other. The integral \( \int_{0}^{2} \int_{0}^{1} x \, dy \, dx \) is a perfect example of this method.
Here's how it works:
  • Inner Integral: We start by integrating \( x \) with respect to \( y \). Since \( x \) doesn't change with \( y \), it acts as a constant.
  • Once the inner integral is computed, we replace it in the double integral, reducing it to a single integral.
  • Outer Integral: Next, we integrate the result of the inner integral with respect to \( x \), finishing the calculation.
The order of integration is usually specified with limits or noted by which variable comes first in the integral sign. This systematic approach helps break down complex integrals into simpler steps.
Antiderivatives
Antiderivatives, also known as indefinite integrals, are foundational in solving integrals. They help us find a function whose derivative is the original function. Our task was to find the antiderivative for the function \( x \).
Here's a step-by-step guide:
  • Identify the function: We focus on finding the antiderivative of \( x \).
  • The antiderivative of \( x \) is \( \frac{x^2}{2} \).
  • Evaluate this antiderivative using limits, substituting the upper limit and subtracting the result of the lower limit.
This process helped us conclude that the antiderivative evaluated from 0 to 2 is \( \frac{4}{2} - 0 = 2 \). Finding antiderivatives is crucial in solving definite integrals, as it allows us to quantify areas under curves.
Calculus Step-by-Step Solutions
Step-by-step solutions in calculus provide a clear path to understanding complex problems. For double integrals, this involves evaluating integrals iteratively.
Our exercise followed these steps:
  • Analyze: Start by assessing the integral structure and the order of integration.
  • Integrate Iteratively: Solve the inner integral first and use its result to simplify the outer integral.
  • Solve the Antiderivative: Find the antiderivative for the function involved and apply the limits.
  • Conclude: After evaluating the limits, derive the final result.
This methodical approach ensures an understanding of each part of the problem. By dividing the problem into manageable sections and addressing each one, complex integrals become manageable and solvable.

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