The parametric equations for the part of the sphere defined by the equation x² + y² + z² = 9, constrained between the planes y = 1 and y = 2, are established as follows: x = 3sin(φ)cos(θ), y = 3cos(φ), and z = 3sin(φ)sin(θ). The bounds for φ are cos(1/3) - 1 < φ < cos(2/3) - 1, while θ ranges from 0 to 2π. This parametrization effectively utilizes the spherical coordinates to describe the specified section of the sphere.
PREREQUISITESStudents studying multivariable calculus, mathematicians working with geometric representations, and educators teaching parametric equations and spherical coordinates.
anubis01 said:Oh okay so I can say z=3cos\phi
and from there plug in the values to find the \phi bound to be cos(1/3)-1<\phi<cos(2/3)-1 and the \theta bound should just be 0< \theta<2\pi
anubis01 said:oh so then phi bound should be 0<\phi<\pi and since y=3cos\theta the bounds for \theta are
cos(1/3)-1<\theta<cos(2/3)-1
anubis01 said:alright i think i got it now. So bound for theta 0<\theta<2\pi
y=3cos\phi the bounds for phi are then
cos(1/3)-1<\phi<cos(2/3)-1
and the parametric equation for x=3sin\phicos\theta
and z=3sin\phisin\theta