Compute the given integral as a limit of Riemann sums, where R is the rectangle [-1,1] x [1,2]. (Hint: use the second provided image of the Riemann sum to solve). The image shows a double integral: \[\int_{0}^{1} \int_{3y}^{3} dx \, dy\]This expression represents a double integral over a specific region in the xy-plane. To solve this, the integration is performed first with respect to \(x\), where \(x\) ranges from \(3y\) to \(3\), and then with respect to \(y\), where \(y\) ranges from \(0\) to \(1\).### Explanation of the Double Integral- **Region of Integration**: The limits of integration for \(x\) go from the line \(x = 3y\) to the line \(x = 3\), and for \(y\), from \(0\) to \(1\). - **Integration Process**: 1. Integrate with respect to \(x\) from \(3y\) to \(3\). 2. Integrate the result with respect to \(y\) over the interval \([0, 1]\).This integral calculates the area of a region bounded by the lines \(x = 3y\), \(x = 3\), \(y = 0\), and \(y = 1\) in the Cartesian plane. The formula shown in the image is:\[\sum_{i=1}^{n} i = \frac{1}{2} n(n+1)\]This is the formula for the sum of the first \( n \) natural numbers. The sum of all integers from 1 to \( n \) is equal to half of \( n \) multiplied by \( n+1 \). This formula helps in quickly calculating the sum without needing to add each individual number manually.

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Answered: Sofydxd 3y

Math

Advanced Math

Sofydxd 3y

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Compute the given integral as a limit of Riemann sums, where R is the rectangle [-1,1] x [1,2]. (Hint: use the second provided image of the Riemann sum to solve).

The image shows a double integral:

\[
\int_{0}^{1} \int_{3y}^{3} dx \, dy
\]

This expression represents a double integral over a specific region in the xy-plane. To solve this, the integration is performed first with respect to \(x\), where \(x\) ranges from \(3y\) to \(3\), and then with respect to \(y\), where \(y\) ranges from \(0\) to \(1\).

### Explanation of the Double Integral
- **Region of Integration**: The limits of integration for \(x\) go from the line \(x = 3y\) to the line \(x = 3\), and for \(y\), from \(0\) to \(1\).
- **Integration Process**:
1. Integrate with respect to \(x\) from \(3y\) to \(3\).
2. Integrate the result with respect to \(y\) over the interval \([0, 1]\).

This integral calculates the area of a region bounded by the lines \(x = 3y\), \(x = 3\), \(y = 0\), and \(y = 1\) in the Cartesian plane.

The formula shown in the image is:

\[
\sum_{i=1}^{n} i = \frac{1}{2} n(n+1)
\]

This is the formula for the sum of the first \( n \) natural numbers. The sum of all integers from 1 to \( n \) is equal to half of \( n \) multiplied by \( n+1 \). This formula helps in quickly calculating the sum without needing to add each individual number manually.

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