An extremely long, solid nonconducting cylinder has a radius $R_0$. The charge density within the cylinder is a function of the distance $R$ from the axis, given by $\rho_{\mathrm{E}}(R)=\rho_0\left(R / R_0\right)^2$. What is the electric field everywhere inside and outside the cylinder (far away from the ends) in terms of $\rho_0$ and $R_0$ ?

If an electron $\left(m=9.1 \times 10^{-31} \mathrm{~kg}\right)$ escaped from the surface of the inner cylinder in Problem 36 with negligible speed, what would be its speed when it reached the outer cylinder?

The volume of a cylinder is

$$600\pi\ cm^3.$$

The radius of a base of the cylinder is 5 cm. What is the height of the cylinder?

Show that the graph of r = 3 cos t i + 3 sin t j + 3 sin tk is an ellipse, and find the lengths of the major and minor axes. [Hint: Show that the graph lies on both a circular cylinder and a plane and use the result in the previous exercise of the earlier section.]

The diameter of a base of a cylinder is twice the height of the cylinder. a. Write a formula in terms of h for the surface area of the cylinder. b. The surface area of the cylinder is 64π square units. Find the radius an d height of the cylinder.

Find the volume of the right prism or right cylinder. Round your answer to two decimal places.

The cylinder's radius is 5 in. and its length is 12 in.

Find the height of a right cylinder with surface area $160 \pi \mathrm{ft}^{2}$ and radius 5 ft.

Find the volume of the right prism or right cylinder. Round your answer to two decimal places.

The cylinder's radius is 3 ft and its height is 4 ft.

Solve. The diameter of the base of a cylinder is 1.8 m. The height of the cylinder is 0.7 m. Find the exact surface area of the cylinder.

Integrate $g(x, y, z)= x^{4} y\left(y^{2}+z^{2}\right)$ over the portion of the cylinder $y^{2}+z^{2}=25$

Find the right cylinder in the classroom. Measure its height and diameter and calculate its base area. What is its volume?

In Exercise given below, use algebra to express the volume of each solid.

Right cylinder; base circumference $=p \pi$

Find the height of a right cylinder with surface area that is $160 \pi \mathrm{ft}^2$ and radius $5 \mathrm{ft}$.

When you use formulas to find values of dependent variables associated with specific values of independent variables, you need to be careful to "read" the directions of the formula the way they are intended. For example, the formula for surface area of a circular cylinder is $A=2 \pi r^2+2 \pi r h$.

a. What is the surface area of a cylinder with radius 5 inches and height 8 inches? How do you know your calculation used the formula correctly?

b. Sketch a cylinder for which the height is the same length as the radius $r$. Show that the surface area of the cylinder is $A=4 \pi r^2$. What algebraic properties did you use in your reasoning?