Calculating the Volume of a Complex Object Using Triple Integrals in Cylindrical Coordinates

The calculation of complex volumes can be achieved through the use of multiple integrals. In particular, the triplet integral is a powerful tool to find the volume of objects defined by a combination of surfaces and curves. This article will guide you through the process of determining the volume of an object defined by the equations:

x^2y^2 3y x^2y^2 6y z radic;x^2y^2 z 0

Setting Up the Problem

The first step is to set up the integral in cylindrical coordinates. Cylindrical coordinates are a convenient choice because they naturally fit the symmetry of the problem. In cylindrical coordinates, the volume element is given by:

V ∫∫∫V dxdydz ∫∫∫V r dφ dr dz

Transformation to Cylindrical Coordinates

Transform the given shapes into cylindrical coordinates:

x^2y^2 3y: This equation becomes r^2 sin^2 φ 3 sin φ, or r_1 3 sin φ (for φ 0) x^2y^2 6y: This equation becomes r^2 sin^2 φ 6 sin φ, or r_2 6 sin φ (for φ 0) z radic;x^2y^2: This equation becomes r r, where φ is the angle in the xy-plane. z 0: This is the xy-plane, which remains the same in cylindrical coordinates.

Setting Up the Integral

The integral is set up as follows:

V ∫0π dφ ∫3 sin φ6 sin φ r dr ∫0r dz

Integrating with respect to z first:

V ∫0π dφ ∫3 sin φ6 sin φ r^2 dr

Evaluating the Integral

Evaluating the integral with respect to r:

V 1/3 ∫0π (6^3 - 3^3) sin^3 φ dφ

Simplifying:

V 54 ∫0π sin^3 φ dφ

Using trigonometric identities, we can simplify sin^3 φ:

sin^3 φ 3/4 sin φ - 1/4 sin 3φ

So the integral becomes:

V 54 [3/4 ∫0π sin φ dφ - 1/4 ∫0π sin 3φ dφ]

Evaluating the integrals:

3/4 ∫0π sin φ dφ 3/4 [-cos φ]_0^π 3/4 [1 - (-1)] 3/4 * 2 3/2

1/4 ∫0π sin 3φ dφ 1/4 [-1/3 cos 3φ]_0^π 1/4 [-1/3 cos 3π - (-1/3 cos 0)] 1/4 [1/3 1/3] 1/6

Therefore:

V 54 [3/2 - 1/6] 54 [9/6 - 1/6] 54 [8/6] 54 * 4/3 72

The volume of the object is 72 cubic units.