"Determine whether the following statements are true and give an explanation or counterexample.


a. A pyramid is a solid of revolution. "

3–6. Setting up arc length integrals Write and simplify, but do not evaluate, an integral with respect to x that gives the length of the following curves on the given interval.

y = 2 cos 3x on [−π,π]

64–68. Shell method Use the shell method to find the volume of the following solids.

A hole of radius r≤R is drilled symmetrically along the axis of a bullet. The bullet is formed by revolving the parabola y = 6(1−x²/R²) about the y-axis, where 0≤x≤R.

9–20. Arc length calculations Find the arc length of the following curves on the given interval.

x = y⁴/4 + 1/8y², for 1≤y≤2

Use the general slicing method to find the volume of the following solids.

The solid whose base is the region bounded by the semicircle y=√1−x^2 and the x-axis, and whose cross sections through the solid perpendicular to the x-axis are squares

"Determine whether the following statements are true and give an explanation or counterexample.

b. The volume of a hemisphere can be computed using the disk method. "

Use calculus to find the volume of a tetrahedron (pyramid with four triangular faces), all of whose edges have length 4.

Find the volume of the solid whose base is the region bounded by the function $f\left(x\right)=\sqrt{9-x^2}$ and the x-axis with square cross sections perpendicular to the x-axis.

Find the volume for a solid whose base is the region between the curve $y=\sqrt{\sin x}$ and the x-axis on the interval from $[0,\pi ]$ and whose cross sections are equilateral triangles with bases parallel to the y-axis.