2 4 2 [²√² (1 – xy²) dady − -1 0
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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Compute the attached double
![### Double Integral Explanation
This image shows a double integral, which is used to calculate the volume under a surface in a given region in the xy-plane.
The expression is:
\[
\int_{-1}^{2} \int_{0}^{4} (1 - xy^2) \, dx \, dy
\]
#### Explanation:
1. **Integrals**:
- The outer integral \(\int_{-1}^{2}\) indicates integration with respect to \(y\), from \(y = -1\) to \(y = 2\).
- The inner integral \(\int_{0}^{4}\) indicates integration with respect to \(x\), from \(x = 0\) to \(x = 4\).
2. **Function**:
- The integrand is \(1 - xy^2\), which is the function we integrate over the specified region. It represents the height of the surface above each point \((x, y)\).
3. **Calculation Process**:
- First, integrate \(1 - xy^2\) with respect to \(x\) from 0 to 4 while keeping \(y\) constant.
- Then, integrate the result with respect to \(y\) from -1 to 2.
This process will give the total volume under the surface defined by \(1 - xy^2\) and above the region in the xy-plane specified by the bounds \([-1, 2]\) for \(y\) and \([0, 4]\) for \(x\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F45df613f-b2a4-49ad-bfd1-57f03a5fa5b6%2Faed4b3a5-409d-4086-8eea-fe7f5cc73547%2Fsdnptn_processed.png&w=3840&q=75)
Transcribed Image Text:### Double Integral Explanation
This image shows a double integral, which is used to calculate the volume under a surface in a given region in the xy-plane.
The expression is:
\[
\int_{-1}^{2} \int_{0}^{4} (1 - xy^2) \, dx \, dy
\]
#### Explanation:
1. **Integrals**:
- The outer integral \(\int_{-1}^{2}\) indicates integration with respect to \(y\), from \(y = -1\) to \(y = 2\).
- The inner integral \(\int_{0}^{4}\) indicates integration with respect to \(x\), from \(x = 0\) to \(x = 4\).
2. **Function**:
- The integrand is \(1 - xy^2\), which is the function we integrate over the specified region. It represents the height of the surface above each point \((x, y)\).
3. **Calculation Process**:
- First, integrate \(1 - xy^2\) with respect to \(x\) from 0 to 4 while keeping \(y\) constant.
- Then, integrate the result with respect to \(y\) from -1 to 2.
This process will give the total volume under the surface defined by \(1 - xy^2\) and above the region in the xy-plane specified by the bounds \([-1, 2]\) for \(y\) and \([0, 4]\) for \(x\).
![The image contains two mathematical formulas related to summations.
1. **Sum of the first n natural numbers:**
\[
\sum_{j=1}^{n} j = \frac{1}{2} n (n + 1)
\]
This formula represents the sum of the integers from 1 to \( n \). It states that the sum can be calculated using the expression \(\frac{1}{2} n (n + 1)\).
2. **Sum of the squares of the first n natural numbers:**
\[
\sum_{j=1}^{n} j^2 = \frac{1}{6} n (n + 1) (2n + 1)
\]
This formula provides the sum of the squares of integers from 1 to \( n \). The formula used here is \(\frac{1}{6} n (n + 1) (2n + 1)\).
These formulas are often used in mathematical analysis and proofs, especially in topics related to arithmetic series and quadratic sums.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F45df613f-b2a4-49ad-bfd1-57f03a5fa5b6%2Faed4b3a5-409d-4086-8eea-fe7f5cc73547%2Fe1q4k3_processed.png&w=3840&q=75)
Transcribed Image Text:The image contains two mathematical formulas related to summations.
1. **Sum of the first n natural numbers:**
\[
\sum_{j=1}^{n} j = \frac{1}{2} n (n + 1)
\]
This formula represents the sum of the integers from 1 to \( n \). It states that the sum can be calculated using the expression \(\frac{1}{2} n (n + 1)\).
2. **Sum of the squares of the first n natural numbers:**
\[
\sum_{j=1}^{n} j^2 = \frac{1}{6} n (n + 1) (2n + 1)
\]
This formula provides the sum of the squares of integers from 1 to \( n \). The formula used here is \(\frac{1}{6} n (n + 1) (2n + 1)\).
These formulas are often used in mathematical analysis and proofs, especially in topics related to arithmetic series and quadratic sums.
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