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

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

Consider the integral expression involving a region R and various functions, including a double integral with limits involving constants and variables. The problem involves multiple parts: sketching the region of integration, rewriting the integral in polar coordinates, calculating the total mass of a semi-circular disc with a density proportional to the radius, and analyzing three surfaces with respect to their volumes and coordinate systems.

Paper For Above instruction

Introduction

The integration of functions over specific regions in the plane or space plays a fundamental role in calculus, particularly in applications involving mass, volume, and geometric properties. This paper systematically addresses the tasks outlined in the assignment, starting from geometric interpretation, coordinate transformations, and extending to volume calculations for solids with specified density or surface boundaries.

Part 1: Understanding and Rewriting the Double Integral

The initial task involves a double integral over a region R, which appears to be defined with limits involving algebraic expressions such as 5𝑦, 64𝑥, and 213. The integral's integrand contains a complex expression, possibly involving a sum like (5𝑦 + 6) multiplied by the square root of a quadratic expression, i.e., √(4−𝑦^2), with limits hinting at bounded regions in the xy-plane.

To interpret and sketch the region of integration (Part 1A), it is crucial to isolate the bounds and understand their geometric meaning. The limits involving √(4−𝑦^2) suggest a semi-circular shape, as √(4−𝑦^2) is the positive half of a circle of radius 2 centered at the origin. The bounds in y from -2 to 2, and corresponding x limits derived from the expressions, point toward a semi-circular region.

Rewriting the integral in polar coordinates (Part 1B) simplifies the process by leveraging the symmetry of the circular bounds. Polar coordinates (r, θ), where x = r cos θ and y = r sin θ, change the bounds accordingly: r ranges from 0 to 2, θ from 0 to π for the upper semi-circle, and the integrand must be correspondingly transformed, incorporating Jacobian determinant r.

Part 2: Mass of a Semi-Circular Disc

The second problem involves calculating the total mass of a semi-circular disc of radius 4, with a density proportional to the distance from the center, i.e., ρ(r) = k * r, where k is a proportionality constant. The density depends solely on the radius, consistent with radial symmetry, making polar coordinates the ideal choice.

The mass is computed as the double integral over the semi-circular region:

\[ M = \int \int_{D} \rho(r) dA \]

which, in polar coordinates, is:

\[ M = \int_{θ=0}^{π} \int_{r=0}^{4} k r \times r dr dθ \]

where the Jacobian r appears from the transformation. The integral calculates the total mass based on the density function.

Part 3: Volume of Solids Defined by Surfaces

The problem moves to three-dimensional analysis, involving two surfaces:

- Surface 1: 𝑍 = √6 - 𝑌^2 - 𝑍^2

- Surface 2: 𝑍 = √(𝑌^2 + 𝑍^2)

The task involves visualizing the solid bounded by these surfaces (Part 3A) and expressing the volume in different coordinate systems.

In Cartesian coordinates (Part 3B), the volume can be expressed as an iterated integral, with limits derived from the intersection points of the surfaces. In polar coordinates (Part 3C), the symmetry suggests a natural transformation where y and z are represented in terms of r and θ, simplifying the volume calculation. Finally, in the order of coordinates (Part 3D), the integral might be expressed as a triple integral with the order z, y, x (or similar), depending on the geometry.

Conclusion

This analysis demonstrates the importance of coordinate transformations in multivariable calculus—particularly integrating over regions exhibiting symmetry. Recognizing geometric shapes like circles and semi-circular discs facilitates simplifying complex integrals via polar coordinates, thereby making calculation more feasible. Moreover, understanding how to set up integrals in different coordinate systems and orders of integration is essential for tackling a variety of volume and mass problems in calculus.

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