Volumes

[ellion71]

I am having difficulty calculating the volume of a solid encolsed between the cylinder x^2+y^2=9 and the planes z=1 and x+z=5.
Any guidance would be appreciated!

 

[galactus]

z varies from z=1 to x = -x+5

y varies from \(\displaystyle x^{2}+y^{2}=9\Rightarrow y=\pm\sqrt{9-x^{2}}\)

and x varies from -3 to 3.

\(\displaystyle \displaystyle \int_{-3}^{3}\int_{-\sqrt{9-x^{2}}}^{\sqrt{9-x^{2}}}\int_{1}^{-x+5} dzdydx\)

 

[HallsofIvy]

z varies from z=1 to x = -x+5

galactus clearly meant z= -x+ 5.

y varies from \(\displaystyle x^{2}+y^{2}=9\Rightarrow y=\pm\sqrt{9-x^{2}}\)

and x varies from -3 to 3.

\(\displaystyle \displaystyle \int_{-3}^{3}\int_{-\sqrt{9-x^{2}}}^{\sqrt{9-x^{2}}}\int_{1}^{-x+5} dzdydx\)