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 xx-axis from t=0t=0 to t=π2:\displaystyle t=\frac{\pi}{2}: x(t)=3costy(t)=4sint.\begin{aligned} x(t) &= 3 \cos t \\ y(t) &= 4 \sin t. \end{aligned}

Find the volume of the three dimensional shape obtained by revolving the curve described by the parametric equations below about the yy-axis from t=1t=1 to t=5:t=5: x(t)=15t2y(t)=6t.\begin{aligned} x(t) &= 15 t^2 \\ y(t) &= 6 t. \end{aligned}

Find the volume of the three dimensional shape obtained by the following equations from r=0r=0 to r=1, r= 1, from t=0t=0 to t=3, t= 3, and u=0 u = 0 to u=2π: u = 2 \pi: x(r,t,u)=3ty(r,t,u)=rt2+6tcosuz(r,t,u)=rt2+6tsinu.\begin{aligned} 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{aligned}

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

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

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