Problem 29 - Calculus Early Transcendentals: Pearson New International Edition

Question

Six surfaces are given below. Without performing any integration, rank the surfaces in order of their surface area from smallest to largest. Hint: There may be some "ties." (a) The paraboloid \(z=x^{2}+y^{2}\) over the region in the first quadrant and inside the circle \(x^{2}+y^{2}=1\) (b) The hyperbolic paraboloid \(z=x^{2}-y^{2}\) over the region in the first quadrant and inside the circle \(x^{2}+y^{2}=1\) (c) The paraboloid \(z=x^{2}+y^{2}\) over the region inside the rectangle with vertices \((0,0),(1,0),(1,1)\), and \((0,1)\) (d) The hyperbolic paraboloid \(z=x^{2}-y^{2}\) over the region inside the rectangle with vertices \((0,0),(1,0),(1,1)\), and \((0,1)\) (e) The paraboloid \(z=x^{2}+y^{2}\) over the region inside the triangle with vertices \((0,0),(1,0)\), and \((0,1)\) (f) The hyperbolic paraboloid \(z=x^{2}-y^{2}\) over the region inside the triangle with vertices \((0,0),(1,0)\), and \((0,1)\)

Step-by-Step Solution

Answer

e \( \approx \) f < a \( \approx \) b < c \( \approx \) d.

Step 1: Identify the Shapes and Domains

Understanding the domain where each surface is defined is crucial. Calculate the base area for each surface to facilitate comparison later:- Paraboloid (a) and hyperbolic paraboloid (b) are inside a circle sector in the first quadrant with radius 1, i.e., area is \( \frac{\pi}{4} \).- Paraboloid (c) and hyperbolic paraboloid (d) are inside a full rectangle (unit square), i.e., area is 1.- Paraboloid (e) and hyperbolic paraboloid (f) are inside a right triangle, i.e., area is \( \frac{1}{2} \).

Step 2: Analyze the Effect of the Surfaces on Area

The shape of the surface itself affects the total area. - Paraboloid: \( z = x^2 + y^2 \). This surface is upward-opening and contributes more area because it curves away from the base plane.- Hyperbolic Paraboloid: \( z = x^2 - y^2 \). This surface has a saddle shape, often contributing less area since parts are positive while others retract (negative).

Step 3: Compare Paraboloids First

Comparing surfaces that have the same type (paraboloid with paraboloid) based on their base areas:- Paraboloid (a) area over the circle segment is \( \frac{\pi}{4} \), whereas paraboloid (c) has an area of 1, and paraboloid (e) has an area of \( \frac{1}{2} \).- So, in terms of areas: Paraboloid (e) < Paraboloid (a) < Paraboloid (c).

Step 4: Compare Hyperbolic Paraboloids

Now, compare the hyperbolic paraboloids similarly by base area:- Hyperbolic paraboloid (f), a saddle over triangle \( \frac{1}{2} \), has less area due to its shape.- The base area of hyperbolic paraboloid (b) is \( \frac{\pi}{4} \), which approximates \(0.785\), and (d) is 1.- So: Hyperbolic Paraboloid (f) < Hyperbolic Paraboloid (b) < Hyperbolic Paraboloid (d).

Step 5: Rank All Surfaces

Combine identified rankings:- Smallest to largest surface areas are: Paraboloid (e) \( \approx \) Hyperbolic Paraboloid (f) < Paraboloid (a) \( \approx \) Hyperbolic Paraboloid (b) < Paraboloid (c) \( \approx \) Hyperbolic Paraboloid (d).- Acknowledge potential ties due to similar base areas: Triangle base areas are smaller than circle segments, which are smaller than rectangles.

Key Concepts

Surface Area Comparison

When comparing the surface areas of different geometric shapes, it's important to consider both the type of the surface and the region it covers. In this exercise, we are looking at two main surface types: paraboloids and hyperbolic paraboloids, each across different geometric regions.

To rank the surfaces without performing integration, we analyze the contribution of each surface type to the area within specific boundaries. Both the shape of the surface and the domain's area play crucial roles.

• Paraboloids tend to contribute more area as they curve upwards, expanding away from the base.
• Hyperbolic Paraboloids have parts that curve both above and below the base line, often resulting in smaller net surface areas.

Understanding these characteristics allows us to assess and rank the surfaces by area from smallest to largest.

Paraboloids

Paraboloids represent surfaces that are described mathematically as quadratic functions of two variables. In this problem, the equation for the paraboloid is given by \( z = x^2 + y^2 \), an upward-opening surface. This equation causes the paraboloid to resemble a shallow bowl, becoming steeper as you move away from the origin.

For each shape this surface covers, the area can vary:

• Over a circular region (like in case (a)), the area inside the circle sector is determined by the radius, here being 1, resulting in a base area of \( \frac{\pi}{4} \).
• Within a rectangular region (case (c)), the base has a unit area of 1.
• A triangular region (case (e)) gives a triangular slice bag of area \( \frac{1}{2} \).

When comparing paraboloids, a larger base typically means a larger surface area. However, the starting base geometry also plays a crucial part in evaluating the total surface area above it.

Hyperbolic Paraboloids

Hyperbolic paraboloids, or saddle shapes, have a distinct mathematical representation: \( z = x^2 - y^2 \). Unlike paraboloids, these surfaces curve up in one direction and down in another, taking on the appearance of a saddle. These shapes are more complex due to their unique structural twist.

This characteristic makes calculating surface areas for hyperbolic paraboloids vary greatly depending on the domain.

• Inside a circle sector as in (b), the region's quarter-circle base offers an area of \( \frac{\pi}{4} \).
• With a unit square base in (d), the overall area becomes 1.
• The triangular slice under (f) has an area of \( \frac{1}{2} \).

Comparison among hyperbolic paraboloids based on domain highlights how their complex structure can result in smaller perceived net areas compared to paraboloids of similar base areas.

Geometric Regions

Here, the geometric regions are crucial for understanding the base areas over which the surfaces extend. They influence both the computed base area and the extent of the surface's exposure:

• Circle Sector: Represented in exercises (a) and (b), this region is defined by a radius of 1, providing a base area of \( \frac{\pi}{4} \).
• Rectangle: Involves exercises (c) and (d), defined as a unit square, achieving a total base area of 1.
• Triangle: Covers exercises (e) and (f), with a right triangular base providing an area of \( \frac{1}{2} \).

These regions allow us to calculate and compare surface areas indirectly since larger base areas tend to produce larger surface areas when the surfaces themselves have similar shapes.

First Quadrant Analysis

The first quadrant analysis involves looking specifically at how these surfaces are positioned within the first quadrant of a Cartesian plane. This restriction ensures that all coordinates are positive or zero, which simplifies many analytical processes and comparisons.

The first quadrant offers simplicity in examining surface interaction with each shape:

• Ensures positive values for both \(x\) and \(y\), reducing the complexity involved in handling different signs.
• Streamlines analysis of each surface area by providing a consistent frame of reference from one shape to the next.

Studying surfaces in this quadrant helps in systematically comparing the contribution of each surface type to the overall area, and is particularly useful in visualizing geometry in calculus contexts.