91Ó°ÊÓ

Give an example of: A triangular region in the plane for which both horizontal and vertical slices work just as easily.

Consider the region bounded by \(y=e^{x}\) the \(x\) -axis, and the lines \(x=0\) and \(x=1 .\) Find the volume of the solid. The solid whose base is the given region and whose cross-sections perpendicular to the \(x\) -axis are squares.

Give an example of: The equation of a circle in polar coordinates.

Consider the region bounded by \(y=e^{x}\) the \(x\) -axis, and the lines \(x=0\) and \(x=1 .\) Find the volume of the solid. The solid whose base is the given region and whose cross-sections perpendicular to the \(x\) -axis are semicircles.

Are the statements true or false? Give an explanation for your answer. The integral \(\int_{-3}^{3} \pi\left(9-x^{2}\right) d x\) represents the volume of a sphere of radius 3.

Give an example of: A polar curve \(r=f(\theta)\) that is symmetric about neither the \(x\) -axis nor the \(y\) -axis.

The region is rotated around the \(y\) -axis. Write, then evaluate, an integral or sum of integrals giving the volume. Bounded by \(y=2 x, y=6-x, y=0\).

Give an example of: A polar curve \(r=f(\theta)\) other than a circle that is symmetric about the \(x\) -axis.

The region is rotated around the \(y\) -axis. Write, then evaluate, an integral or sum of integrals giving the volume. Bounded by \(y=2 x, y=6-x, x=0\).

Are the statements true or false? Give an explanation for your answer. The integral \(\int_{0}^{r} \pi \sqrt{r^{2}-y^{2}} d y\) gives the volume of a hemisphere of radius \(r\).