Compute the volume of the column with a square base R= {(x,y): |x|< 3, |y| <3} cut by the plane z = 10 - x- y.

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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**Title: Calculating the Volume of a Column with a Square Base**

**Text:**

**Problem Statement:**

Compute the volume of the column with a square base \( R = \{ (x, y) : |x| \leq 3, \ |y| \leq 3 \} \) cut by the plane \( z = 10 - x - y \).

**Analysis:**

- **Square Base**: The base \( R \) is defined as a square in the xy-plane, with sides parallel to the axes, extending from \(-3\) to \(3\) for both \(x\) and \(y\).
- **Plane Equation**: The plane cutting the column has the equation \( z = 10 - x - y \), sloping downwards as \(x\) and \(y\) increase.
  
To find the volume, consider integrating the height (given by the plane equation) over the region \( R \).

**Volume Formula:**

The volume \( V \) is calculated by evaluating the double integral:

\[
V = \int_{-3}^{3} \int_{-3}^{3} (10 - x - y) \, dy \, dx
\]

This setup involves integrating \( z = 10 - x - y \) over the square base \( R = \{ (x, y) : |x| \leq 3, \ |y| \leq 3 \} \).

**Steps for Calculation:**

1. **Outer Integral (dx)**: Integrate over \( x \) from \(-3\) to \(3\).
   
2. **Inner Integral (dy)**: For each fixed \( x \), integrate over \( y \) from \(-3\) to \(3\).

This approach utilizes properties of double integrals to find the volume of the column defined by the parameters above.
Transcribed Image Text:**Title: Calculating the Volume of a Column with a Square Base** **Text:** **Problem Statement:** Compute the volume of the column with a square base \( R = \{ (x, y) : |x| \leq 3, \ |y| \leq 3 \} \) cut by the plane \( z = 10 - x - y \). **Analysis:** - **Square Base**: The base \( R \) is defined as a square in the xy-plane, with sides parallel to the axes, extending from \(-3\) to \(3\) for both \(x\) and \(y\). - **Plane Equation**: The plane cutting the column has the equation \( z = 10 - x - y \), sloping downwards as \(x\) and \(y\) increase. To find the volume, consider integrating the height (given by the plane equation) over the region \( R \). **Volume Formula:** The volume \( V \) is calculated by evaluating the double integral: \[ V = \int_{-3}^{3} \int_{-3}^{3} (10 - x - y) \, dy \, dx \] This setup involves integrating \( z = 10 - x - y \) over the square base \( R = \{ (x, y) : |x| \leq 3, \ |y| \leq 3 \} \). **Steps for Calculation:** 1. **Outer Integral (dx)**: Integrate over \( x \) from \(-3\) to \(3\). 2. **Inner Integral (dy)**: For each fixed \( x \), integrate over \( y \) from \(-3\) to \(3\). This approach utilizes properties of double integrals to find the volume of the column defined by the parameters above.
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