Asked by Anonymous
Find the volume of a pyramid with height h and rectangular base with dimensions b and 2b.
Essentially, I'm trying to derive the volume of a pyramid using integrals and a graph and revolving the plane about an axis. I hope that's kind of clear and you guys understand what I'm asking.
Thank you very much
Essentially, I'm trying to derive the volume of a pyramid using integrals and a graph and revolving the plane about an axis. I hope that's kind of clear and you guys understand what I'm asking.
Thank you very much
Answers
Answered by
bobpursley
That seems to me to be the hard way to integrate.
take a small rectangular box dV somewhere along the height.
dV= dh*darea
darea= dw dl=(H-h)(b)*(H-h)2b /H^2
dV=1/H^2 (H-h)^2 2b^2 dh h from 0 to H
V= 2b^2/H^2 INT (H-h)^2 dh= 2b^2 /3 H^2 *(H-h)^3 from zero to H
=2b^2/3 H
which you knew, the volume of any rectangular prism is 1/3 area base*height
take a small rectangular box dV somewhere along the height.
dV= dh*darea
darea= dw dl=(H-h)(b)*(H-h)2b /H^2
dV=1/H^2 (H-h)^2 2b^2 dh h from 0 to H
V= 2b^2/H^2 INT (H-h)^2 dh= 2b^2 /3 H^2 *(H-h)^3 from zero to H
=2b^2/3 H
which you knew, the volume of any rectangular prism is 1/3 area base*height
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