Using triple integral, find the volume of the solid bounded by the cylinder y^2+z^2=1 and planes x=0 and x+y+z=2...how to draw the graph?and how to do the calculation?
-
Best to change it to polar:
y=rcos(t)
z= rsin(t)
dV = r dr dt dx
The limits on r will be 0 to 1,
t from 0 to 2 pi
and x from 0 to 2-r(sin(t) + cos(t))
===> ∫∫∫ r dx dr dt
The picture is the cylinder 1 unit radius around the x-axis going from the yz plane to the plane x+y+z=2... To draw this plane, it's easiest to start with the intercepts... (2,0,0), (0,2,0), and (0,0,2)...
For the normal configuration with x out of the page, and y horizontal right, and z up, then you have a cylinder coming toward you and a plane going down at a diagonal as it come toward you and the little to the right that comes from the pic...
Perhaps easiest is to draw the normal vector first: C*<1, 1, 1>
See if slader has your book
y=rcos(t)
z= rsin(t)
dV = r dr dt dx
The limits on r will be 0 to 1,
t from 0 to 2 pi
and x from 0 to 2-r(sin(t) + cos(t))
===> ∫∫∫ r dx dr dt
The picture is the cylinder 1 unit radius around the x-axis going from the yz plane to the plane x+y+z=2... To draw this plane, it's easiest to start with the intercepts... (2,0,0), (0,2,0), and (0,0,2)...
For the normal configuration with x out of the page, and y horizontal right, and z up, then you have a cylinder coming toward you and a plane going down at a diagonal as it come toward you and the little to the right that comes from the pic...
Perhaps easiest is to draw the normal vector first: C*<1, 1, 1>
See if slader has your book