Calculus

# 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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