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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.

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

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

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