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Calculating the Volume Between a Surface and a Cylinder: A Comprehensive Guide
Calculating the Volume Between a Surface and a Cylinder: A Comprehensive Guide
Understanding complex geometrical shapes and their volumes, especially when combined with constraints, is essential in advanced mathematics and engineering. This article provides a detailed step-by-step guide on calculating the volume between a specific surface and a cylinder, utilizing cylindrical coordinates for the integration process. We will also explore the mathematical symbols and equations, ensuring clarity and precision in the calculations.
Introduction
Volume calculation between a surface and a constrained shape, such as a cylinder, is a critical concept in mathematics and engineering. This tutorial will focus on the specific problem where the surface is defined by the equation x^2 y^2 3z^2, and the cylinder is defined by x^2 y^2 4y. The volume is calculated using cylindrical coordinates and a carefully set-up integral. The final result will be presented, ensuring a clear understanding of the process involved.
Understanding the Cylinder
The equation of the cylinder is given by:
x^2 y^2 4y
Our first step is to rewrite this equation to better understand the geometry:
x^2 y^2 - 4y 0
Completing the square for the y term, we get:
x^2 (y - 2)^2 4
This shows that the cylinder is centered at (0, 2) in the xy-plane with a radius of 2. This transformation is crucial in setting up the volume integral.
Setting Up the Volume Integral
The volume V can be found by integrating the z value, which is derived from the surface equation x^2 y^2 3z^2, over the region defined by the cylinder. In cylindrical coordinates, where x r cos θ and y r sin θ, the equation for z is:
z sqrt{(r^2 / 3)}
The volume element in cylindrical coordinates is dV r dz dr dθ. We will integrate z from 0 to r / sqrt{3}, r from 0 to 2, and θ from 0 to 2π.
Determining the Limits of Integration
The limits of integration for z are from 0 to r / sqrt{3}. For θ, the angle ranges from 0 to 2π. For r, it is more complex, as the region is not a simple circle. Let's transform the cylinder equation into cylindrical coordinates:
r^2 4 sin^2 θ - 4 sin θ
This means r ranges from 0 to 2 sin θ due to the trigonometric nature of the constraint.
Volume Integral
The volume integral becomes:
V ∫_0^{2π} ∫_0^{2 sin θ} ∫_0^{r / sqrt{3}} r dz dr dθ
First, integrating with respect to z:
∫_0^{r / sqrt{3}} r dz r · z|0^{r / sqrt{3}} r · (r / sqrt{3}) r^2 / sqrt{3}
Now, integrating with respect to r:
V ∫_0^{2π} ∫_0^{2 sin θ} (r^2 / sqrt{3}) dr dθ
∫_0^{2π} (1 / sqrt{3}) · [r^3 / 3]_0^{2 sin θ} dθ
∫_0^{2π} (1 / sqrt{3}) · (8 sin^3 θ / 3) dθ
(8 / (3 sqrt{3})) ∫_0^{2π} sin^3 θ dθ
Using the identity sin^3 θ sin θ - sin θ cos^2 θ and integration by substitution, we can find:
∫_0^{2π} sin^3 θ dθ 4/3
Substituting this result back into the volume integral:
V (8 / (3 sqrt{3})) · (4 / 3) (32 / (9 sqrt{3}))
Final Result: The volume between the surface x^2 y^2 3z^2 and the xy-plane constrained by the cylinder x^2 y^2 4y is:
boxed{32 / (9 sqrt{3})}
Conclusion
This detailed calculation showcases the power of cylindrical coordinates and integration in solving complex geometrical problems. By breaking down the problem into manageable steps, we have arrived at a precise result, enhancing our understanding of the mathematical relationships involved.