Question Type 5: Finding the volume of revolution of a region about the y axis Exercises

Find the volume of the solid formed by revolving the region under $y = x^3$ from $x=0$ to $x=1$ about the $y$-axis.

Calculate the volume generated by revolving the region under the curve $y = x^2$ from $x=0$ to $x=2$ about the $y$-axis.

Find the volume of the solid generated by revolving the region between $y=1/x$ and the $x$-axis from $x=1$ to $x=2$ about the $y$-axis.

Find the volume of the solid obtained by revolving the region bounded by $y = x$, $x = 3$, and $y = 0$ about the $y$-axis.

Determine the volume of the solid generated by revolving the region under $y=\sqrt{x}$ from $x=0$ to $x=9$ about the $y$-axis.

Compute the volume of the solid formed by revolving the region between $y=x$ and $y=x^2$ for $0\le x\le1$ about the $y$-axis.

Compute the volume of the solid obtained by revolving the region between $x = y^2$ and $x = 1$ for $0\le y\le 1$ about the $y$-axis.

Find the volume of the solid formed by revolving the region bounded by $y = 4 - x^2$ and $y = 0$ from $x=-2$ to $x=2$ about the $y$-axis.

Compute the volume of the solid obtained by revolving the region under $y=\ln(x)$ from $x=1$ to $x=e$ about the $y$-axis.

Compute the volume of the solid obtained by revolving the region bounded by $y=\cos x$, the $x$-axis, from $x=0$ to $x=\tfrac{\pi}{2}$ about the $y$-axis.

Determine the volume of the solid obtained by revolving the region bounded by $x=y^2$, $x=y+2$, for $0\le y\le2$ about the $y$-axis.

Find the volume of the solid generated by revolving the region under $y=e^{-x}$ from $x=0$ to $x=\infty$ about the $y$-axis.