>	Maple Project #2

>	Pan, Xiaodong

>	with(plots):

Warning, the name changecoords has been redefined

>	plot3d(1+1/5*sin(theta)*sin(6*phi), theta=0..2*Pi, phi=0..Pi, coords=spherical);

>	Int(Int(Int(1*sin(phi)*rho^2, rho=0..1+1/5*sin(theta)*sin(6*phi)), theta=0..2*Pi), phi=0..Pi);

>

value(%);

>	plot3d(1+1/5*sin(11*theta)*sin(6*phi), theta=0..2*Pi, phi=0..Pi, coords=spherical);

>	Int(Int(Int(1*sin(phi)*rho^2, rho=0..1+1/5*sin(11*theta)*sin(6*phi)), theta=0..2*Pi), phi=0..Pi);

>

value(%);

>	plot3d(1+1/5*sin(12*theta)*sin(6*phi), theta=0..2*Pi, phi=0..Pi, coords=spherical);

>	Int(Int(Int(1*sin(phi)*rho^2, rho=0..1+1/5*sin(12*theta)*sin(6*phi)), theta=0..2*Pi), phi=0..Pi);

>

value(%);

>	plot3d(1+1/5*sin(13*theta)*sin(6*phi), theta=0..2*Pi, phi=0..Pi, coords=spherical);

>	Int(Int(Int(1*sin(phi)*rho^2, rho=0..1+1/5*sin(13*theta)*sin(6*phi)), theta=0..2*Pi), phi=0..Pi);

>

value(%);

>	plot3d(1+1/5*sin(23*theta)*sin(6*phi), theta=0..2*Pi, phi=0..Pi, coords=spherical);

>	plot3d(1+1/5*sin(24*theta)*sin(6*phi), theta=0..2*Pi, phi=0..Pi, coords=spherical);

>	plot3d(1+1/5*sin(25*theta)*sin(6*phi), theta=0..2*Pi, phi=0..Pi, coords=spherical);

>	plot3d(1+1/5*sin(35*theta)*sin(6*phi), theta=0..2*Pi, phi=0..Pi, coords=spherical);

>	plot3d(1+1/5*sin(36*theta)*sin(6*phi), theta=0..2*Pi, phi=0..Pi, coords=spherical);

>	plot3d(1+1/5*sin(37*theta)*sin(6*phi), theta=0..2*Pi, phi=0..Pi, coords=spherical);

>	plot3d(1+1/5*sin(47*theta)*sin(6*phi), theta=0..2*Pi, phi=0..Pi, coords=spherical);

>	plot3d(1+1/5*sin(48*theta)*sin(6*phi), theta=0..2*Pi, phi=0..Pi, coords=spherical);

When the coefficient of theta changes, the shape of the graph changes. However, the volune of it does not change. when the coefficent of theta is very small, such as one, the graph is like a ball that is oppressed by our feet. when the coefficient  of theta gets larger, but less than twelve, the shape of the graph is like a pineapple. when the coefficient of theta reaches twelve, the shape of it becomes a sphere, and the shape of sphere repeats itself whenever the coefficient of the theta is the multiple of twelve.  The two shapes that close to twelve or a multiple of twelve are the same. For example, the shapes of the graphs are the same when the coefficient of the theta is eleven and thirteen. It is just like a sine curve that all shapes are repeating themselves.

>	plot3d(1+1/5*sin(5*theta)*sin(phi), theta=0..2*Pi, phi=0..Pi, coords=spherical);

>	Int(Int(Int(1*sin(phi)*rho^2, rho=0..1+1/5*sin(5*theta)*sin(phi)), theta=0..2*Pi), phi=0..Pi);

>

value(%);

>	plot3d(1+1/5*sin(5*theta)*sin(11*phi), theta=0..2*Pi, phi=0..Pi, coords=spherical);

>	Int(Int(Int(1*sin(phi)*rho^2, rho=0..1+1/5*sin(5*theta)*sin(11*phi)), theta=0..2*Pi), phi=0..Pi);

>

value(%);

>	plot3d(1+1/5*sin(5*theta)*sin(12*phi), theta=0..2*Pi, phi=0..Pi, coords=spherical);

>	Int(Int(Int(1*sin(phi)*rho^2, rho=0..1+1/5*sin(5*theta)*sin(12*phi)), theta=0..2*Pi), phi=0..Pi);

>

value(%);

>	plot3d(1+1/5*sin(5*theta)*sin(23*phi), theta=0..2*Pi, phi=0..Pi, coords=spherical);

>	Int(Int(Int(1*sin(phi)*rho^2, rho=0..1+1/5*sin(5*theta)*sin(23*phi)), theta=0..2*Pi), phi=0..Pi);

>

value(%);

>	plot3d(1+1/5*sin(5*theta)*sin(24*phi), theta=0..2*Pi, phi=0..Pi, coords=spherical);

>	Int(Int(Int(1*sin(phi)*rho^2, rho=0..1+1/5*sin(5*theta)*sin(24*phi)), theta=0..2*Pi), phi=0..Pi);

>

value(%);

>	plot3d(1+1/5*sin(5*theta)*sin(25*phi), theta=0..2*Pi, phi=0..Pi, coords=spherical);

>	Int(Int(Int(1*sin(phi)*rho^2, rho=0..1+1/5*sin(5*theta)*sin(25*phi)), theta=0..2*Pi), phi=0..Pi);

>

value(%);

>	plot3d(1+1/5*sin(5*theta)*sin(47*phi), theta=0..2*Pi, phi=0..Pi, coords=spherical);

>	Int(Int(Int(1*sin(phi)*rho^2, rho=0..1+1/5*sin(5*theta)*sin(47*phi)), theta=0..2*Pi), phi=0..Pi);

>

value(%);

>	plot3d(1+1/5*sin(5*theta)*sin(48*phi), theta=0..2*Pi, phi=0..Pi, coords=spherical);

>	Int(Int(Int(1*sin(phi)*rho^2, rho=0..1+1/5*sin(5*theta)*sin(48*phi)), theta=0..2*Pi), phi=0..Pi);

>

value(%);

>	plot3d(1+1/5*sin(5*theta)*sin(49*phi), theta=0..2*Pi, phi=0..Pi, coords=spherical);

>	Int(Int(Int(1*sin(phi)*rho^2, rho=0..1+1/5*sin(5*theta)*sin(49*phi)), theta=0..2*Pi), phi=0..Pi);

>

value(%);

When the coefficient of phi changes, both the shapes and volumes change. When phi is very small, such as one, the shape of it is like a starfruit that is different from when the coefficient of theta is one. When the coefficient of phi gets closer to twenty-four, the shpe of it likes a durian. When the coefficient of phi is twenty-four, the shape of the graph is a sphere, and the shapes repeat themselves whenever the coefficient of phi is the multiple of twenty-four. The shapes of the graphs are vey similar when the coefficient is either a litter larger or smaller than twenty-four. The shapes are repeating themselves, but their volumes are different. The volumes gets smaller as the coefficient of phi gets larger in the interval of one to infinite. The volumes get larger when the coefficient of phi gets larger in the interval of zero to one, but not including zero and one.

>	plot3d(1+1/5*sin(5*theta)*sin((1/2)*phi), theta=0..2*Pi, phi=0..Pi, coords=spherical);

>	Int(Int(Int(1*sin(phi)*rho^2, rho=0..1+1/5*sin(5*theta)*sin((1/2)*phi)), theta=0..2*Pi), phi=0..Pi);

>

value(%);

>	plot3d(1+1/5*sin(5*theta)*sin((1/3)*phi), theta=0..2*Pi, phi=0..Pi, coords=spherical);

>	Int(Int(Int(1*sin(phi)*rho^2, rho=0..1+1/5*sin(5*theta)*sin((1/3)*phi)), theta=0..2*Pi), phi=0..Pi);

>

value(%);

>	plot3d(1+1/5*sin(5*theta)*sin((1/4)*phi), theta=0..2*Pi, phi=0..Pi, coords=spherical);

>	Int(Int(Int(1*sin(phi)*rho^2, rho=0..1+1/5*sin(5*theta)*sin((1/4)*phi)), theta=0..2*Pi), phi=0..Pi);

>

value(%);