Page 2 Of 31: Consider Fxy DA 𝑅 5y 64x213

Page 2 Of 31 2 Points Consider 𝑓𝑥𝑦𝑑𝐴 𝑅 5𝑦 64𝑥213

Consider the integral involving the region defined by the inequalities and the Cartesian or polar coordinate transformations provided. The task involves sketching the region described by the integral, rewriting the integral in a different coordinate system (polar coordinates), setting up an integral to determine the total mass of a semi-circular disc with a specific mass density, and expressing the volume of a given solid both in Cartesian and polar coordinates with specified coordinate orders. The substance of the problem revolves around understanding the geometric regions, understanding the conversion between coordinate systems, and applying integral calculus to compute areas, masses, and volumes relevant to the given surfaces and solids.

Paper For Above instruction

Introduction

Integral calculus plays a fundamental role in understanding and computing geometric quantities such as areas, volumes, and masses. This paper focuses on multiple interconnected problems involving regions in the plane and solids in space, primarily using Cartesian and polar coordinate systems. The goal is to develop a comprehensive understanding of how to sketch regions, convert integrals between systems, and formulate integrals for calculation purposes, specifically related to a semi-circular disc and certain defined solids.

Part 1: Sketching and Changing Coordinates of a Region

The first problem involves analyzing an integral over a region in the plane defined by the inequalities: x and y variables combined with constraints like 5y, 6, and powers of x. The initial step involves understanding the region R over which the integral is calculated and sketching this region accurately.

To sketch the region, it is necessary to interpret the inequalities that form the boundaries. For example, inequalities involving 5y and 6 define horizontal bands or bounds, whereas inequalities involving powers of x, such as 64x^2 or similar, describe parabolic or elliptical boundaries. We can identify the intersection points by solving the equations obtained by setting the boundary expressions equal to each other.

Transforming the integral into polar coordinates involves converting Cartesian elements: x = r cos θ, y = r sin θ, with the Jacobian for the transformation, r dr dθ. Given the purpose is to rewrite the integral in the order of dθ dr or dr dθ, the limits must be adjusted based on the region's bounds in polar coordinates.

Specifically, this process often simplifies the integration by exploiting symmetry or easier boundary expressions in polar form, such as circles or sectors.

Part 2: Mass Computation of a Semi-Circular Disc

The second problem pertains to calculating the total mass of a thin semi-circular disc of radius 4, where the mass density is proportional to the distance from the center.

The density function can be expressed as ρ(r) = k r, where k is a proportionality constant, and r ranges from 0 to 4 within the semi-circle. The area element in polar coordinates, dA = r dr dθ, allows setting up the integral for the total mass M as:

M = ∫∫_semi-circle ρ(r) dA = ∫_θ=0^{π} ∫_r=0^{4} k r * r dr dθ = k ∫₀^{π} dθ ∫₀^{4} r^2 dr.

Evaluating this integral involves integrating r^2 from 0 to 4 and θ from 0 to π, then multiplying by the constant k. The resulting total mass gives insight into how density distribution affects the total mass in bounded regions.

Part 3: Volume of a Solid with Defined Surfaces

Surface 1 and Surface 2

Given two surfaces, 𝑆1: z = √6 - y^2 - x^2 and 𝑆2: z = √(y^2 + x^2), the solid 𝑆 is bounded between these surfaces. Sketching the solid involves understanding the intersection of these surfaces in three-dimensional space and visualizing the bounded region.

Expressing the volume in Cartesian coordinates in the order (x, y, z), we integrate over the appropriate bounds defined by the surfaces for z, then over the x and y bounds determined by the intersection curves.

Switching to polar coordinates simplifies the process because of the radial symmetry: x = r cos θ, y = r sin θ, and the surfaces can be rewritten accordingly to facilitate easier integration. The order of integration impacts the limits, which depend on the intersection curves of the two surfaces. In polar coordinates, the volume integral involves integrating r from 0 to the boundary determined by the surfaces and θ over the relevant angular sector, typically 0 to 2π.

Discussion and Implications

Understanding the process of sketching regions and transforming integrals between coordinate systems is vital for efficient computation in calculus and applied mathematics. Converting integral limits and surfaces into a more manageable coordinate setup allows precise evaluation of areas, volumes, and mass distributions. These techniques extend into various fields, including physics, engineering, and economics, where spatial and resource modeling are essential.

Conclusion

The problems discussed demonstrate the importance of coordinate transformations, geometric intuition, and integral calculus in solving complex spatial problems. Mastery of sketching regions and changing coordinate systems enhances problem-solving efficiency and accuracy, enabling precise calculations of physical quantities like mass and volume. These skills remain fundamental for advanced applications across scientific and engineering domains.

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