14.3 Volume Integrals

1.Evaluate where and denotes the closed region bounded by the planes

      , , , .

http://matrix.skku.ac.kr/cal-lab/cal-14-5-1.html

 Sol)

 

x,y,z=var('x,y,z');

integral(integral(integral(45*x^2*y, z, 0, 8-4*x-2*y), y, 0, 4-2*x), x, 0, 2)

  128


2. Evaluate where and is the surface whose sides are given cylinder , where bottom is the disk in the plane , and whose top is the part of the plane that lies above .

http://math2.skku.ac.kr/home/pub/55 


 (Sol)

x,y,z,theta=var('x,y,z,t')

p1 = implicit_plot3d(z==1+x, (x,-2,2), (y, -2,2), (z, 0,2), opacity=0.2, color="red", mesh=True);

p2 = implicit_plot3d(x^2+y^2==1, (x,-2,2), (y, -2,2), (z, 0,2), opacity=0.3, color="blue", mesh=True);

p3 = plot3d(0, (x,-2,2), (y, -2,2), opacity=0.3, color="orange", mesh=True);

show(p1+p2+p3, aspect_ratio=1)


 


x = cos(t);

y = sin(t);

z = z;

assume (0 <= t <= 2*pi());

assume (0 <= z <= 1+x);

i,j,k=var('i,j,k');

R=matrix(SR, [[i,j,k], [diff(x,t),diff(y,t),diff(z,t)], [diff(x,z),diff(y,z),diff(z,z)]]).determinant();

R

  i*cos(t) + j*sin(t)


vector_R=vector(SR, [cos(t),sin(t)]);

vector_R.norm()

  sqrt(abs(sin(t))^2 + abs(cos(t))^2)


integral(integral(z^2*1,z,0,1+cos(theta)),theta,0,2*pi())

  5/3*pi