Use cylindric coordinates to calculate the volume of the solid.
The solid is the form determined by the paraboloid x=6-(y^2)-(z^2) and the cone x=sqrt((y^2)+(z^2)).
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You need to investigate the intersection of paraboloid and cone. Notice that `y^2 + z^2 = r^2` , hence:
`x = 6-r^2`
`x=sqrt(r^2) =gt x=r`
You need to set the equations above equal such that:
`6-r^2 = r =gt r^2 + r - 6 = 0`
`r_(1,2) = (-1+-sqrt(1+24))/2 =gt r_1 = (-1+5)/2 =gt r_1 = 2`
`r_2 = -3`
Hence, you may evaluate the volume using triple integral such that:
`V = int_0^(2pi) theta dtheta int_0^2 rdr int_(1-r^2)^r dx`
You need to evaluate the inner integral `int_(1-r^2)^r dx ` such that:
`int_(1-r^2)^r dx = x|_(1-r^2)^r`
`int_(1-r^2)^r dx = r -1 + r^2`
You need to evaluate the middle integral `int_0^2 r(r -1 +r^2)dr ` such that:
`int_0^2 r(r -1 + r^2)dr = int_0^2 r^2dr - int_0^2 rdr+ int_0^2 r^3 dr`
`int_0^2 r(r -1 + r^2)dr = (r^3/3 - r^2/2+ r^4/4)|_0^2`
`int_0^2 r(r -1 + r^2)dr = (8/3 - 4/2 + 16/4)`
`int_0^2 r(r -1 + r^2)dr = 14/3`
Hence, `int_0^(2pi) theta d theta int_0^2 r(r -1 + r^2)dr = 2pi*14/3.`
Hence, evaluating the volume of solid using cylindrical coordinates yields `V = 28pi/3` .
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