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Magazine Chapter 5: Problem 10
Evaluate the double integrals where \(R\) is the rectangle \([0,2] \times[-1,0].\) $$\iint_{R}\left(|y| \cos \frac{1}{4} \pi x\right) d y d x$$
Short Answer
Expert verified
The value is \(\frac{2}{\pi}\).
Step by step solution
01
Understand the Region of Integration
The region \( R \) is a rectangle defined by the Cartesian product \([0,2] \times [-1,0]\). This means \( x \) varies from 0 to 2, and \( y \) varies from -1 to 0.
02
Setup the Double Integral
Set up the double integral for the function \(|y| \, \cos \left(\frac{1}{4} \pi x\right)\). Since \(|y|\) is the absolute value of \(y\), it is equal to \(-y\) in the interval \([-1,0]\). Thus, the integral becomes:\[\int_{0}^{2} \int_{-1}^{0} (-y) \cos \left(\frac{1}{4} \pi x\right) \ dy \ dx\]
03
Integrate with Respect to y
Integrate \(-y \cos \left(\frac{1}{4} \pi x\right)\) with respect to \(y\). Since \(\cos \left(\frac{1}{4} \pi x\right)\) is constant with respect to \(y\), treat it as a constant:\[\int_{-1}^{0} (-y) \cos \left(\frac{1}{4} \pi x\right) \ dy = \cos \left(\frac{1}{4} \pi x\right) \int_{-1}^{0} -y \, dy\]
04
Compute the Inner Integral
Calculate the integral \(\int_{-1}^{0} -y \, dy\):\[\int -y \, dy = -\frac{y^2}{2} \Bigg|_{-1}^{0} = 0 - \left( -\frac{1}{2} \right) = \frac{1}{2}\]
05
Integrate with Respect to x
Now integrate the result of the inner integral, \(\frac{1}{2} \cos \left(\frac{1}{4} \pi x\right)\), over \([0,2]\):\[\int_{0}^{2} \frac{1}{2} \cos \left(\frac{1}{4} \pi x\right) \, dx = \frac{1}{2} \left[ \frac{4}{\pi} \sin \left(\frac{1}{4} \pi x\right) \right]_{0}^{2}\]
06
Compute the Final Integral
Evaluate the last expression:\[\frac{1}{2} \cdot \frac{4}{\pi} \left[ \sin \left(\frac{1}{2} \pi \right) - \sin(0) \right] = \frac{2}{\pi} \cdot 1 = \frac{2}{\pi}\]
07
Conclusion
The value of the integral over the rectangle \( R \) is \(\frac{2}{\pi}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Region of Integration
In double integration, the **region of integration** is an area over which we want to calculate the integral of a function. It gives us the limits within which we evaluate the function.
For this exercise, the region of integration, denoted as \( R \), is a rectangle described by the Cartesian product \([0,2] \times [-1,0]\). This definition tells us how the variables \( x \) and \( y \) change.
For this exercise, the region of integration, denoted as \( R \), is a rectangle described by the Cartesian product \([0,2] \times [-1,0]\). This definition tells us how the variables \( x \) and \( y \) change.
- \( x \) varies from 0 to 2, which means all x-coordinates of the rectangle lie within this interval.
- \( y \) varies from -1 to 0, making all y-coordinates fall within this specified range.
Absolute Value in Integration
The **absolute value** function is used to ensure that any given quantity is non-negative, regardless of the input's original sign.
In our exercise, we dealt with the function \(|y| \cos \left( \frac{1}{4} \pi x \right)\) under double integration over the region \( R \). Here, the absolute value plays an essential role in accurately computing integrals across intervals that include negative numbers.
In our exercise, we dealt with the function \(|y| \cos \left( \frac{1}{4} \pi x \right)\) under double integration over the region \( R \). Here, the absolute value plays an essential role in accurately computing integrals across intervals that include negative numbers.
- For the interval \([-1,0]\), \(|y|\) is equal to \(-y\), since \(y\) is negative within this range, effectively converting negative values to their positive counterparts.
- This simplification allows us to work with data effectively by using standard mathematical operations without concern for varying signs within the specified range.
Trigonometric Integration
**Trigonometric integration** deals with integrals that include trigonometric functions such as sine or cosine. These functions provide periodic, wave-like outputs that can add complexity to integral calculations.
In the given exercise, the cosine function \( \cos \left( \frac{1}{4} \pi x \right) \) appears in our integral. When integrating this function with respect to \( x \), it鈥檚 important to remember the rules for the antiderivative of cosine.
In the given exercise, the cosine function \( \cos \left( \frac{1}{4} \pi x \right) \) appears in our integral. When integrating this function with respect to \( x \), it鈥檚 important to remember the rules for the antiderivative of cosine.
- The integral of \( \cos(kx) \) with respect to \( x \) is \( \frac{1}{k} \sin(kx) + C \), where \( k \) is constant.
- In our case, \( k = \frac{1}{4} \pi \), resulting in an antiderivative that uses these specific constants.
