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.
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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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8735dee1-ec31-4346-b70c-e4d10f64e04c%2F93df5fa7-4216-4c57-9839-6095ab231576%2F6s7mtt4.png&w=3840&q=75)
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