Find the volume formed by rotating the region under \(y=x\) from \(x=0\) to \(x=2\) about the \(x\)-axis.

Find the volume formed by rotating the region under \(y=\sqrt{x}\) from \(x=0\) to \(x=4\) about the \(x\)-axis.

Find the volume formed by rotating the region between \(y=2\) and \(y=x\) for \(0 \le x \le 2\) about the \(x\)-axis.

Find the volume formed by rotating the region under \(y=x^2\) from \(x=0\) to \(x=1\) about the \(y\)-axis. Use cylindrical shells.

A solid has base bounded by \(y=4-x^2\) and the \(x\)-axis. Cross-sections perpendicular to the \(x\)-axis are squares. Find the volume.

Find the volume formed by rotating the region between \(y=x\) and \(y=x^2\) about the \(x\)-axis.

Find the volume formed by rotating the region between \(y=x\) and \(y=x^2\) about the \(y\)-axis. Use cylindrical shells.

Find the volume formed by rotating the region bounded by \(x=y^2\), \(y=0\), and \(y=2\) about the \(y\)-axis.

Find the volume formed by rotating the region between \(y=x\), \(y=0\), \(x=0\), and \(x=1\) about the horizontal line \(y=-1\).

A learner rotates the region under \(y=x\) from \(x=0\) to \(x=3\) about the \(x\)-axis and computes \(\pi\int_0^3 x\,dx=9\pi/2\). Explain the error and find the correct volume.