Problem 35 - Thomas Calculus

Question

The region cut from the solid elliptical cylinder \(x^{2}+4 y^{2} \leq 4\) by the \(x y\) -plane and the plane \(z=x+2\)

Step-by-Step Solution

Answer

The region is a 3D volume between the elliptical cylinder, the xy-plane, and the tilted plane z=x+2.

Step 1: Understand the Problem

We have an elliptical cylinder defined by the inequality \(x^2 + 4y^2 \leq 4\) along with two planes: \(xy\)-plane (\(z=0\)) and \(z=x+2\). Our task is to find the region formed by the intersection of these surfaces.

Step 2: Describe the Cylinder

The elliptical cylinder is oriented along the \(z\)-axis with cross-sections that are ellipses given by \(x^2 + 4y^2 = 4\). For all \(z\), these ellipses are constant, making the cylinder infinite in the \(z\)-direction.

Step 3: Identify Planes

The \(xy\)-plane is the plane where \(z = 0\), and the plane \(z = x + 2\) is tilted so that \(z\) increases linearly with \(x\), starting from 2 when \(x=0\).

Step 4: Intersection with the XY-Plane

At the \(xy\)-plane, \(z=0\). Substitute this into the plane equation \(z=x+2\), giving the line \(0 = x + 2\) or \(x = -2\). This line intersects the elliptical cylinder, providing part of the boundary of the cut region.

Step 5: Intersection with the Plane z = x + 2

For the plane \(z = x + 2\), consider where it cuts through the cylinder. Along this plane, the elliptical region is bound by substituting \(z = x + 2\) as a plane cutting through. We thus have three boundaries: the surface of the elliptical cylinder, the \(xy\)-plane, and the plane \(z = x + 2\).

Step 6: Describe the Region

The region of interest is a three-dimensional form that is cut off at \(z=0\) from below and above by \(z = x + 2\), slicing an elliptical shape along where these planes intersect the cylinder.

Key Concepts

Elliptical Cylinder

An elliptical cylinder is a fascinating three-dimensional structure characterized by its distinct cross-section, which is an ellipse. In mathematics, it can be described using an inequality or an equation. In this case, the elliptical cylinder is defined by the inequality \(x^2 + 4y^2 \leq 4\). This inequality reveals a crucial property of the elliptical cylinder: its cross-sections in every plane parallel to the xy-plane are ellipses of the same size and shape.

• The cylinder is oriented along the z-axis, implying that for any value of z, the cross-section will always be an identical ellipse.
• The equation \(x^2 + 4y^2 = 4\) describes the boundary of these ellipses.

The cylinder is 'infinite' in nature along the z-axis as it stretches endlessly in both the positive and negative z-direction.

Intersection of Planes

The intersection of planes involves determining where two or more planes meet. In our scenario, we have two critical planes: the xy-plane and the tilted plane described by the equation \(z = x + 2\).

• The xy-plane represents the horizontal plane where z is held constant at zero. That is, \(z = 0\).
• The plane \(z = x + 2\) is sloped, indicating z changes linearly with x. When x is zero, z starts at a value of 2, meaning the plane begins its path two units above the xy-plane.

Equating these planes helps us find their intersection points, essential for understanding the boundaries of the region formed.

Three-Dimensional Region

Such a region is formed by the intersection of the elliptical cylinder with the two planes. Three-dimensional regions can be complex as they include elements from multiple intersecting surfaces.

• The region is cut from above by the plane \(z = x + 2\) and from below by the xy-plane.
• Since the elliptical cylinder is infinitely extended along z, the planes slice through it to limit its extent in three-dimensional space.

Visualizing this can be tricky, but it can be interpreted as a segment of the infinite cylinder enclosed between two flat surfaces.

Geometric Boundaries

Geometric boundaries define the limits of shapes and forms. Here, the boundaries of the three-dimensional region are determined by the surfaces that intersect.

• The elliptical boundary is dictated by the elliptical cylinder's shape, specifically \(x^2 + 4y^2 = 4\).
• One side is capped by the xy-plane at \(z=0\), which effectively truncates the elliptical cylinder at its base.
• The other side is capped by the plane \(z = x + 2\), giving the cut region a sloping surface.

These boundaries help delineate the specific segment of space contained within the overlap of the cylinder and the two planes.