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

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

Calculate the volume of the solid obtained by revolving the region under $y = \ln x$ from $x = 1$ to $x = e$ 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.

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.

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

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

Find the volume of the solid generated by revolving the region between $y = \frac{1}{x}$ and the $x$-axis from $x = 1$ to $x = 2$ 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.

Find 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.

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.