Problem 32 - Thomas Calculus

Question

The region cut from the cylinder \(x^{2}+y^{2}=4\) by the plane \(z=0\) and the plane \(x+z=3\) (GRAPH NOT COPY)

Step-by-Step Solution

Answer

A slanted triangular region is formed where the inclined plane cuts through the cylinder in the first quadrant.

Step 1: Understanding the Geometrical Setup

We are given a cylinder centered along the z-axis with radius 2, as represented by the equation \( x^2 + y^2 = 4 \). The plane \( z = 0 \) is the horizontal plane that intersects the base of the cylinder. The plane \( x + z = 3 \) is a slanted plane cutting through the cylinder.

Step 2: Determine the Cylinder's Boundaries

The equation of the cylinder \( x^2 + y^2 = 4 \) implies that the maximum extent of both \( x \) and \( y \) is from -2 to 2. This confines our region within a circle of radius 2 in the \( xy-plane \).

Step 3: Intersect Plane with Cylinder

The plane \( x + z = 3 \) can be rearranged to find \( z = 3 - x \). For the intersection of this plane and the cylinder, we substitute \( z = 3 - x \) into the cylindrical constraints. The intersection only happens where both the cylinder and the plane equations are satisfied.

Step 4: Analyze the Intersection Constraints

For the range where the plane intersects the cylinder, we plug \( z = 3 - x \) into the equation of the cylinder (\( x^2 + y^2 \leq 4 \)). Setting \( x \) using the equation \( x = 3 - z \), we should find the values of \( x \) for which the plane exists within the cylinder's radius. This determines the extent (\( -2 \leq x \leq 2 \)), and thus, \( 0 \leq z \leq 3 \).

Step 5: Calculate Region of Intersection

To find the exact region cut by these boundaries: Project onto the xy-plane for \( z = 0 \) (circular area) and then use the equation \( z = 3 - x \) to ensure that the slanted plane fits within this circle by verifying \( x^2 + y^2 \leq 4 \) whenever \( x = [1,-1] \).By setting maximum and minimum values (with respect to the cylinder boundary \( x^2+y^2=4 \) and within the considered region intersection), the slanted plane will intersect these boundaries forming a semi-triangular shape in the xy-slice.

Step 6: Conclusion

The intersection region is a slice of the cylindrical side cut by the inclined plane \( z = 3 - x \). The projection is a bounded semi-region of a circle from \(-2 \leq x \leq 2\) height limit from \(z=0\) up to \(z=3-x\).

Key Concepts

Cylinder Intersection

The concept of cylinder intersection involves understanding how different geometric shapes can cut through a cylinder. In this exercise, we have a cylinder centered along the z-axis with a radius of 2 described by the equation \( x^2 + y^2 = 4 \). This equation defines a three-dimensional structure extending infinitely along the z-axis. Two planes intersect this cylinder: the horizontal plane \( z = 0 \) and the slanted plane \( x + z = 3 \). These planes intersect with the cylinder to form a unique geometric region.

• The horizontal plane \( z = 0 \) slices the cylinder at its base, establishing a circle with a radius of 2 on the xy-plane.
• The slanted plane \( x + z = 3 \) introduces more complexity as it cuts diagonally through the cylinder. This plane can be rearranged to \( z = 3 - x \) and ultimately affects the shape of the intersection volume.

To determine the intersection accurately, we combine these equations to find where both cylinder and plane conditions meet, thus helping us visualize the resulting sections.

Geometric Integration

Geometric integration is a technique that allows us to calculate properties like volume, area, and other measurements for complex shapes formed by geometric intersections. This approach involves breaking down the region of interest into more manageable parts that can be calculated using integration methods.In this task, integration is not performed explicitly, but understanding the geometric breakdown helps set the stage for it. For the cylinder and intersecting planes, consider:

• The initial cylinder provides a circular boundary in the xy-plane within the limits of \( x^2 + y^2 = 4 \).
• The varying region within the cylinder defined by the plane \( x + z = 3 \) alters this circular section into a more dynamic intersection zone as the plane changes.

Understanding this combines geometry with calculus, allowing for precise calculations of volumes or areas resulting from such complex intersections.

3D Coordinate Geometry

3D coordinate geometry is a powerful tool in calculus for visualizing and solving problems involving multiple dimensions. In three-dimensional space, we use coordinates \((x, y, z)\) to specify positions, and equations to describe geometric objects like planes, cylinders, and spheres.For this exercise, 3D coordinate geometry helps us:

• Describe the cylinder using the equation \( x^2 + y^2 = 4 \), defining a cylindrical space extending indefinitely along the z-axis.
• Characterize the horizontal plane with \( z = 0 \) and interpret the slanted plane \( x + z = 3 \) as slicing through the cylinder.
• Determine the region of intersection in terms of points satisfying both the cylinder and plane equations.

These concepts allow us to visualize how these geometrical elements interact within the 3D space, and by analyzing the intersection, extract further geometric insights.