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Chapter 29: Problem 26

Evaluate each of the given double integrals. $$\int_{1}^{2} \int_{0}^{\pi / 4} r \sec ^{2} \theta d \theta d r$$

Short Answer

Expert verified
The value of the double integral is \( \frac{3}{2} \).

Step by step solution

01

Identify the Integration Order

The given double integral is \( \int_{1}^{2} \int_{0}^{\pi / 4} r \sec ^{2} \theta \, d \theta \, d r \). We need to integrate with respect to \( \theta \) first, and then with respect to \( r \).
02

Inner Integration with Respect to \( \theta \)

Calculate the inner integral \( \int_{0}^{\pi/4} \sec^2 \theta \, d\theta \). The integral of \( \sec^2 \theta \) with respect to \( \theta \) is \( \tan \theta \). So, \[ \int_{0}^{\pi/4} \sec^2 \theta \, d\theta = \left[ \tan \theta \right]_0^{\pi/4} = \tan(\pi/4) - \tan(0) = 1 - 0 = 1. \]
03

Substitute Inner Integral Result

Substitute the result of the inner integral back into the outer integral:\[ \int_{1}^{2} r \cdot 1 \, d r = \int_{1}^{2} r \, d r. \]
04

Outer Integration with Respect to \( r \)

Integrate \( r \) with respect to \( r \):\[ \int r \, d r = \frac{1}{2} r^2 + C. \] Using the limits from 1 to 2:\[ \left[ \frac{1}{2} r^2 \right]_1^2 = \frac{1}{2} (2^2) - \frac{1}{2} (1^2) = \frac{1}{2} (4) - \frac{1}{2} (1) = 2 - \frac{1}{2} = \frac{3}{2}. \]

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Key Concepts

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

Integration Order
Understanding the integration order is crucial when working with double integrals. It tells us which variable to integrate first. In the expression \( \int_{1}^{2} \int_{0}^{\pi / 4} r \sec ^{2} \theta \, d \theta \, d r \), the integration order specifies that \( \theta \) is the inner variable and should be integrated first, followed by \( r \).
  • Inner integration: Perform the integration with respect to \( \theta \).
  • Outer integration: Perform the integration with respect to \( r \).
Considering the correct order ensures that we address the dependencies between the variables and maintain the integrity of the solution.
Inner Integration
Inner integration involves dealing with the first variable in the integration order, which, in this case, is \( \theta \). The process here is to evaluate the integral \( \int_{0}^{\pi/4} \sec^2 \theta \, d\theta \). To solve this, we recognize a common integral: the derivative of \( \tan \theta \) is \( \sec^2 \theta \). So,\[ \int_{0}^{\pi/4} \sec^2 \theta \, d\theta = \left[ \tan \theta \right]_0^{\pi/4}. \]Evaluating this, we find:\[ \tan(\pi/4) - \tan(0) = 1 - 0 = 1. \]This result of 1 is then used in the outer integration to simplify the calculations.
Outer Integration
Once the inner integration is complete, we substitute the result into the outer integral. Now, the task is to evaluate \( \int_{1}^{2} r \, d r \). The function becomes simpler as a result of the inner integration.Recognizing that the integral of \( r \) is a standard form:\[ \int r \, d r = \frac{1}{2} r^2 + C. \]Applying the limits from 1 to 2, we calculate:\[ \left[ \frac{1}{2} r^2 \right]_1^2 = \frac{1}{2} (2^2) - \frac{1}{2} (1^2) = 2 - \frac{1}{2} = \frac{3}{2}. \]This final result of \( \frac{3}{2} \) is the answer to the original double integral, achieved by executing the outer integration properly.
Calculus Problems
Calculus problems involving double integrals can seem daunting at first, but they become manageable with practice and understanding of key concepts such as integration order. The integration order guides us through which variable to integrate first and simplifies the computation.
  • Breaking down the problem: Start by separating the integral into inner and outer parts.
  • Simplifying: Use known integral formulas to handle each part.
  • Solution assembly: Combine the results from each integration step to final solution.
By methodically approaching each variable and handling it in the integration process, calculus problems become more predictable and less confusing. Effective solutions depend on methodical work and attention to integration order.

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Most popular questions from this chapter

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