Volume

of a region in space

This worksheet investigates the volume of the region of space under the graph of  

and above the graph of z = 0 with x between -2.5 and 2.5 and y between -0.5 and 1.5.

Here is a picture of the region of space (or solid) whose volume we are going to approximate.

>	with(plots):plot3d(9-x^2-y^2,x=-2.5..2.5,y=-0.5..1.5,view=0..10,axes=boxed,orientation=[100,60],filled=true);

>

Here we are drawing the top surface over the region of the xy-coordinate plane described above and filling it in with 10 boxes.

>	part1:=plot3d(4,x=1.5..2.5,y=0.5..1.5,axes=boxed,orientation=[100,60],filled=true):

>	part2:=plot3d(7,x=0.5..1.5,y=0.5..1.5,axes=boxed,orientation=[100,60],filled=true):

>	part3:=plot3d(8,x=-0.5..0.5,y=0.5..1.5,axes=boxed,orientation=[100,60],filled=true):

>	part4:=plot3d(7,x=-1.5..-0.5,y=0.5..1.5,axes=boxed,orientation=[100,60],filled=true):

>	part5:=plot3d(4,x=-2.5..-1.5,y=0.5..1.5,axes=boxed,orientation=[100,60],filled=true):

>	part6:=plot3d(5,x=1.5..2.5,y=-0.5..0.5,axes=boxed,orientation=[100,60],filled=true):

>	part7:=plot3d(8,x=0.5..1.5,y=-0.5..0.5,axes=boxed,orientation=[100,60],filled=true):

>	part8:=plot3d(9,x=-0.5..0.5,y=-0.5..0.5,axes=boxed,orientation=[100,60],filled=true):

>	part9:=plot3d(8,x=-1.5..-0.5,y=-0.5..0.5,axes=boxed,orientation=[100,60],filled=true):

>	part10:=plot3d(5,x=-2.5..-1.5,y=-0.5..0.5,axes=boxed,orientation=[100,60],filled=true):

>	surface:=plot3d(9-y^2-x^2,x=-2.5..2.5,y=-0.5..1.5,view=0..10,axes=boxed,orientation=[100,60]):

>	display(part1,part2,part3,part4,part5,part6,part7,part8,part9,part10,surface);

Here is the code for generating all the pictures in the animation.  I kept adding more elements to the "display" command at the bottom to generate each successive picture in the animated giff and then I used the mouse to change the viewing angle to generate more pictures to use in the animation.

>	B1:=spacecurve([2.5,-0.5+t,0,t=0..2],color=blue,thickness=3):

>	B2:=spacecurve([1.5,-0.5+t,0,t=0..2],color=blue,thickness=3):

>	B3:=spacecurve([0.5,-0.5+t,0,t=0..2],color=blue,thickness=3):

>	B4:=spacecurve([-0.5,-0.5+t,0,t=0..2],color=blue,thickness=3):

>	B5:=spacecurve([-1.5,-0.5+t,0,t=0..2],color=blue,thickness=3):

>	B6:=spacecurve([-2.5,-0.5+t,0,t=0..2],color=blue,thickness=3):

>	B7:=spacecurve([-2.5+t,-0.5,0,t=0..5],color=blue,thickness=3):

>	B8:=spacecurve([-2.5+t,0.5,0,t=0..5],color=blue,thickness=3):

>	B9:=spacecurve([-2.5+t,1.5,0,t=0..5],color=blue,thickness=3):

>	H6:=spacecurve([2,1,t,t=0..4],color=red,thickness=3):

>	H7:=spacecurve([1,1,t,t=0..7],color=red,thickness=3):

>	H8:=spacecurve([0,1,t,t=0..8],color=red,thickness=3):

>	H9:=spacecurve([-1,1,t,t=0..7],color=red,thickness=3):

>	H10:=spacecurve([-2,1,t,t=0..4],color=red,thickness=3):

>	H11:=spacecurve([2,0,t,t=0..5],color=red,thickness=3):

>	H12:=spacecurve([1,0,t,t=0..8],color=red,thickness=3):

>	H13:=spacecurve([0,0,t,t=0..9],color=red,thickness=3):

>	H14:=spacecurve([-1,0,t,t=0..8],color=red,thickness=3):

>	H15:=spacecurve([-2,0,t,t=0..5],color=red,thickness=3):

>	surface:=plot3d(9-y^2-x^2,x=-2.5..2.5,y=-0.5..1.5,view=0..10,axes=boxed,orientation=[-100,70]):

>	part1:=plot3d(4,x=1.5..2.5,y=0.5..1.5,axes=boxed,orientation=[-100,70],filled=true):

>	part2:=plot3d(7,x=0.5..1.5,y=0.5..1.5,axes=boxed,orientation=[-100,70],filled=true):

>	part3:=plot3d(8,x=-0.5..0.5,y=0.5..1.5,axes=boxed,orientation=[-100,70],filled=true):

>	part4:=plot3d(7,x=-1.5..-0.5,y=0.5..1.5,axes=boxed,orientation=[-100,70],filled=true):

>	part5:=plot3d(4,x=-2.5..-1.5,y=0.5..1.5,axes=boxed,orientation=[-100,70],filled=true):

>	part6:=plot3d(5,x=1.5..2.5,y=-0.5..0.5,axes=boxed,orientation=[-100,70],filled=true):

>	part7:=plot3d(8,x=0.5..1.5,y=-0.5..0.5,axes=boxed,orientation=[-100,70],filled=true):

>	part8:=plot3d(9,x=-0.5..0.5,y=-0.5..0.5,axes=boxed,orientation=[-100,70],filled=true):

>	part9:=plot3d(8,x=-1.5..-0.5,y=-0.5..0.5,axes=boxed,orientation=[-100,70],filled=true):

>	part10:=plot3d(5,x=-2.5..-1.5,y=-0.5..0.5,axes=boxed,orientation=[-100,70],filled=true):

>	display(B1,B2,B3,B4,B5,B6,B7,B8,B9,surface);

>	display(B1,B2,B3,B4,B5,B6,B7,B8,B9,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,surface,part1,part2,part3,part4,part5,part6,part7,part8,part9,part10);

Computing the volume of the region using an iterated double integral

>	int(int(9-x^2-y^2,y=-1/2..3/2),x=-5/2..5/2);

Approximating the volume of the region from 10 approximating rectangular boxes (see picture above)

In this case the area of the base of each approximating box is one.

>	Yvals:=-1:

>	Volume:=0:

>	for i from 1 to 2 do   Yvals:=Yvals+1;   Xvals:=-3;   for j from 1 to 5 do     Xvals:=Xvals+1;     Volume:=Volume+9-Xvals^2-Yvals^2;   end do: end do:

>

Volume;

Approximating the volume of the region from 40 approximating rectangular boxes

In this case the area of the base of each approximating box is 1/4.

>	Yvals:=-0.75:

>	Volume:=0:

>	for i from 1 to 4 do   Yvals:=Yvals+0.5;   Xvals:=-2.75;   for j from 1 to 10 do     Xvals:=Xvals+0.5;     Volume:=Volume+(9-Xvals^2-Yvals^2)/4;   end do: end do:

>

Volume;

Approximating the volume of the region from 160 approximating rectangular boxes

In this case the area of the base of each approximating box is 1/16.

>	Yvals:=-0.625:

>	Volume:=0:

>	for i from 1 to 8 do   Yvals:=Yvals+0.25;   Xvals:=-2.625;   for j from 1 to 20 do     Xvals:=Xvals+0.25;     Volume:=Volume+(9-Xvals^2-Yvals^2)/16;   end do: end do:

>

Volume;

Approximating the volume of the region from 640 approximating rectangular boxes

In this case the area of the base of each approximating box is 1/64.

>	Yvals:=-0.5625:

>	Volume:=0:

>	for i from 1 to 16 do   Yvals:=Yvals+0.125;   Xvals:=-2.5625;   for j from 1 to 40 do     Xvals:=Xvals+0.125;     Volume:=Volume+(9-Xvals^2-Yvals^2)/64;   end do: end do:

>

Volume;

>