Volume: Method of cylindrical shells

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Find the volume of the solid generated by revolving the region bounded by the curves $y=\sqrt{x}, \, \, x=2$ and $y=0$ about the y-axis.  solution

Let R be the region bounded by the curves $y=\sqrt[3]{x}, \, \, y=0$ and $x=1$. Find the volume of the solid generated by revolving the region R about the y-axis.  solution

Find the volume of the solid obtained by rotating the region bounded by the curves $y=2x$ and $y=x^2$ about the $y$-axis. solution

Find the volume of the solid generated by revolving the region bounded by the curves $y=\sqrt{x}, \,\, y=2$ and $x=0$ about the x-axis.  solution

Find the volume of the solid obtained by rotating about the line $x = -2$, the region bounded by $y=x$ and $y=x^2$.   solution

Find the volume of the solid obtained by rotating about the line $x = -1$, the region bounded by $y=2x^2-x^3$ and $y=0$.   solution

Find the volume of the solid generated by revolving the region bounded by the curves $y=\sqrt{x}, \,\, x=4$ and $y=0$ about the the line $y=-2$.  solution

Find the volume of the solid obtained by rotating about the line $x = 2$, the region bounded by $y=x^4, \,\, y=0$ and $x=1$.   solution