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Chapter 15: Problem 35

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

Short Answer

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

Step by step solution

01

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.
02

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.
03

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 \(x=0\).
04

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.
05

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 \(z = x + 2\).
06

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

How would you evaluate the double integral of a continuous function \(f(x, y)\) over the region \(R\) in the \(x y\) -plane enclosed by the triangle with vertices \((0,1),(2,0),\) and \((1,2) ?\) Give reasons for your answer.

Evaluate the integral $$ \int_{0}^{\infty} \frac{e^{-a x}-e^{-b x}}{x} d x $$ (Hint: Use the relation $$ \frac{e^{-a x}-e^{-b x}}{x}=\int_{a}^{b} e^{-x y} d y $$ to form a double integral and evaluate the integral by changing the order of integration.)

(a) find the spherical coordinate limits for the integral that calculates the volume of the given solid and (b) then evaluate the integral. The solid bounded below by the sphere \(\rho=2 \cos \phi\) and above by the cone \(z=\sqrt{x^{2}+y^{2}}\)

Unbounded region Prove that $$ \begin{aligned} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-x^{2}-y^{2}} d x d y &=\lim _{b \rightarrow \infty} \int_{-b}^{b} \int_{-b}^{b} e^{-x^{2}-y^{2}} d x d y \\ &=4\left(\int_{0}^{\infty} e^{-x^{2}} d x\right)^{2} \end{aligned} $$

Use a CAS double-integral evaluator to find the integrals in Exercises \(711-76 .\) Then reverse the order of integration and evaluate, again with a CAS. $$ \int_{0}^{2} \int_{0}^{4-y^{2}} e^{x y} d x d y $$
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