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Use cylindric coordinates to calculate the volume of the solid.The solid is the form...

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bhanbhan | Student, Undergraduate | eNoter

Posted March 17, 2012 at 4:28 AM via web

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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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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted March 17, 2012 at 6:34 PM (Answer #1)

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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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