The region bounded by $y=x^2$, $y=0$, and $x=3$ is rotated about the $y$-axis. Which shell integral gives the volume?

The region under $y=4-x$ on $[0,4]$ is rotated about the $y$-axis. Which shell integral gives the volume?

The region bounded by $y=\sqrt{x}$, $y=0$, and $x=9$ is rotated about the $y$-axis. Which shell integral gives the volume?

The region bounded by $y=2x+1$, $y=0$, $x=0$, and $x=2$ is rotated about the $y$-axis. Which integral uses shells?

The first-quadrant region bounded by $y=9-x^2$, $y=0$, and $x=0$ is rotated about the $y$-axis. Which shell integral gives the volume?

The region between $y=\cos x$ and the $x$-axis on $\left[0,\frac{\pi }{2}\right]$ is rotated about the $y$-axis. Which shell integral gives the volume?

The region bounded by $y=\sqrt{x}$, $y=0$, and $x=4$ is rotated about $x=6$. Which shell integral gives the volume?

The region bounded by $y=x^2+1$, $y=0$, $x=0$, and $x=2$ is rotated about $x=5$. Which shell integral gives the volume?

The region bounded by $y=3x$, $y=0$, and $x=2$ is rotated about $x=-1$. Which shell integral gives the volume?

The first-quadrant region bounded by $y=4-x^2$, $y=0$, and $x=0$ is rotated about $x=3$. Which shell integral gives the volume?

The region bounded by $y=\frac{1}{x}$, $y=0$, $x=1$, and $x=3$ is rotated about $x=4$. Which shell integral gives the volume?

The region bounded by $x=y^2$ and $x=4$ is rotated about the $x$-axis. Using the upper half, which shell integral gives the volume?

The first-quadrant region bounded by $x=0$, $x=9-y^2$, and $y=0$ is rotated about the $x$-axis. Which shell integral gives the volume?

The region bounded by $x=y+1$, $x=5$, $y=0$, and $y=4$ is rotated about the $x$-axis. Which shell integral gives the volume?

The region between $x=y^2$ and $x=2y$ for $0\le y\le 2$ is rotated about the $x$-axis. Which shell integral gives the volume?

Find the exact volume when the region bounded by $y=x$, $y=0$, and $x=2$ is rotated about the $y$-axis.

Find the exact volume when the region bounded by $y=x^2$, $y=0$, and $x=2$ is rotated about the $y$-axis.

Find the exact volume when the region under $y=4-x$ on $[0,4]$ is rotated about the $y$-axis.

Find the exact volume when the region bounded by $y=\sqrt{x}$, $y=0$, and $x=4$ is rotated about the $y$-axis.

Find the exact volume when the region bounded by $y=x^2$, $y=0$, $x=0$, and $x=1$ is rotated about $x=2$.

The region between $y=\sin x$ and the $x$-axis on $[0,\pi ]$ is rotated about the $y$-axis. Why might shells be preferable to washers?

For the region under $y=x^2$ on $[0,3]$ rotated about the $y$-axis, what are the radius and height of a vertical shell?

For the region under $y=\sqrt{x}$ on $[0,4]$ rotated about $x=6$, what are the radius and height of a vertical shell?

For the region bounded by $x=y^2$ and $x=4$ with $0\le y\le 2$ rotated about the $x$-axis, what are a horizontal shell's radius and height?