Question Type 2: Volumes of revolution Exercises

Find the volume of the solid formed by revolving the region bounded by the curve $y=\sqrt{x}$, the $x$-axis, and the line $x=4$ about the $y$-axis.

Find the volume of the solid produced by revolving the region between $y=x^2$ and $y=2$ from $x=0$ to $x=\sqrt{2}$ about the line $y=3$.

Find the volume of the solid generated by revolving the region bounded by $y=\ln x$, the $x$-axis, $x=1$, and $x=e$ about the $y$-axis.

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

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

Find the volume of the solid obtained by revolving the curve $y=\frac{1}{x}$ from $x=1$ to $x=4$ about the $x$-axis. [4]

Find the volume of the solid generated by revolving the region bounded by $x=y^2$ and $x=y+2$ for $-1\le y\le2$ about the $y$-axis.

Calculate the volume of the solid produced by revolving the region between $y=x$ and $y=\sqrt{x}$ from $x=0$ to $x=1$ about the $y$-axis.

Calculate the volume obtained by revolving the region bounded by $y=x^3$, $y=0$, and $x=2$ about the $y$-axis.

Compute the volume of the solid obtained by revolving the region under $y=e^x$ from $x=0$ to $x=1$ about the $y$-axis.

Find the volume of the solid formed by revolving the curve $y=\cos x$ from $x=0$ to $x=\frac{\pi}{2}$ about the $x$-axis.

Calculate the volume of the solid generated by revolving the region bounded by $y=\ln x$, the $x$-axis, $x=1$, and $x=e$ about the $x$-axis.