# plot and volume of y^2+z^2=1 cut by the plane y=x and x=1 in the first quadrant

I need to plot and find the volume of y^2+z^2=1 cut by the plane y=x and x=1 in the first quadrant

So far,

RegionPlot3D[
y^2 + z^2 - 1 <= 1, {x, 0, 1}, {y, 0, x}, {z, 0, Sqrt[1 - y^2]},
AxesLabel -> Automatic]
RegionPlot3D[
y <= x && x <= 1 && z <= Sqrt[1 - y^2], {x, 0, 1}, {y, 0, x}, {z, 0,
Sqrt[1 - y^2]}, AxesLabel -> Automatic]


And for volume

NIntegrate[Sqrt[1 - y^2], {x, 0, 1}, {y, 0, x}]


Both produce an output I'm sure is incorrect. Thank you for any help.

• Funny title---plot and volume---what about solve and area, integrate and length :) ? Nov 13 '21 at 18:36

Since the finite region in the first quadrant of x-y plane is 0 <= x <= 1, 0 <= y <= x,and the finite region in the space is as below.

reg=ImplicitRegion[{y^2 + z^2 <= 1, 0 <= x <= 1, 0 <= y <= x}, {x, y,
z}]
reg// Volume
reg//Region


1/6 (-4 + 3 Pi)

• RegionPlot3D[{x, y, z} \[Element] reg, AxesLabel -> Automatic] doesnt seem to work Nov 13 '21 at 0:27
• @calcNoob1 Thanks, RegionPlot3D[reg,AxesLabel -> Automatic] also work. Nov 13 '21 at 0:29
• it worked thank you very much :) Nov 13 '21 at 0:31