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Calculus

Calculus of Parametric Equations

Parametric Equations - Volume

         

Find the volume of the three dimensional shape obtained by revolving the curve described by the parametric equations below about the \(x\)-axis from \(t=0\) to \(\displaystyle t=\frac{\pi}{2}:\) \[\begin{align} x(t) &= 3 \cos t \\ y(t) &= 4 \sin t. \end{align}\]

Find the volume of the three dimensional shape obtained by revolving the curve described by the parametric equations below about the \(y\)-axis from \(t=1\) to \(t=5:\) \[\begin{align} x(t) &= 15 t^2 \\ y(t) &= 6 t. \end{align}\]

Find the volume of the three dimensional shape obtained by the following equations from \(r=0\) to \( r= 1, \) from \(t=0\) to \( t= 3, \) and \( u = 0 \) to \( u = 2 \pi: \) \[\begin{align} x(r,t,u) &= 3 t \\ y(r,t,u) &= r\sqrt{ t^2+ 6t } \cos u \\ z(r,t,u) &= r\sqrt{ t^2 + 6t } \sin u. \end{align}\]

Find the volume of the three dimensional shape obtained by the following equations from \(t=-5\) to \( t= 5, \) \(u=-4\) to \( u= 4, \) and \( w = 0 \) to \( w = 1: \) \[\begin{align} x(t,u,w) &= (1-w)t \\ y(t,u,w) &= (1-w)u \\ z(t,u,w) &= 9 w. \end{align}\]

Find the volume of the three dimensional shape obtained by the following equations from \( r = 0 \) to \( r = 1, \) from \(t=0\) to \( t=3 \pi ,\) and \( u = 0 \) to \( u = 2 \pi: \) \[\begin{align} x(r,t,u) &= 9 rt ( \cos u ) \\ y(r,t,u) &= 2 rt ( \sin u) \\ z(r,t,u) &= 4t. \end{align}\]

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