Question

evaluate the integral by changing to spherical coordinates. 6 0 √ 36 − x 2 0 √ 72 − x 2 − y 2 √ x 2 y2 xy dz dy dx

Answer 1

The student's question pertains to converting a triple integral from cartesian to spherical coordinates for easier evaluation, involving the use of radial distance, polar, and azimuthal angles.

Step-by-step explanation:

The student's question involves evaluating a triple integral by converting to spherical coordinates. This is a common technique used in advanced mathematics, particularly in multiple integration, to simplify the integration over a spherical domain. In spherical coordinates, any point in space is represented by three variables: the radial distance r from the origin, the polar angle θ (theta) with respect to the z-axis, and the azimuthal angle φ (phi) with respect to the x-axis. The relationship between cartesian and spherical coordinates is given by x = r sin(θ) cos(φ), y = r sin(θ) sin(φ), and z = r cos(θ). For the integral provided, one would first need to define the limits of integration for r, θ, and φ that correspond to the given cartesian bounds. Then, the determinant of the Jacobian matrix of the transformation is used to convert dx dy dz to spherical coordinates, which is r^2 sin(θ). The final step is to express the integrand in terms of r, θ, and φ and evaluate the integral over the new limits.

Answer 2

The question is about converting a triple integral from Cartesian to spherical coordinates for easier evaluation, a common technique in higher-level mathematics.

Step-by-step explanation:

The student's question pertains to the evaluation of a triple integral using spherical coordinates in the field of Mathematics. The integral involves a complex region of integration that is more naturally expressed in spherical coordinates. In mathematics, spherical coordinates (r, θ, φ) relate to Cartesian coordinates (x, y, z) by the formulas x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ. For the integration, one typically needs to identify the bounds for r, θ, and φ that correspond to the given limits in Cartesian coordinates. Then, the function to be integrated and the differential volume element in spherical coordinates, which is r^2 sin θ dr dθ dφ, are used to reformulate the integral.