Determine which value best approximates the volume of the solid generated by revolving the region bounded by the graphs of the equations about the y-axis. (Make your selection on the basis of a sketch of the solid and not by performing any calculations.) $y=2 e^{-x}, y=0, x=0, x=2$ (a) $\frac{3}{2}$, (b) $-2$, (c) $4$, (d) $7.5$, (e) $15$

Use the Theorem of Pappus to find the volume of the solid of revolution. The solid formed by revolving the region bounded by the graphs of y = 2√x-2, y = 0, and x = 6 about the y-axis.

Find the volume of the solid of intersection (the solid common to both) of the two right circular cylinders of radius r whose axes meet at right angles.

Use integration to find the lateral surface area of a right circular cone of height 4 and radius 3.

Use the integration capabilities of a graphing utility to approximate the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis. y= e^(-x²), y = 0, x = 0, x = 2

Give a geometric argument that explains why the integrals have equal values. $\pi \int_{1}^{5}(x-1) d x=2 \pi \int_{0}^{2} y\left[5-\left(y^{2}+1\right)\right] d y$

Write and evaluate the definite integral that represents the area of the surface generated by revolving the curve on the indicated interval about the x-axis. $y=\sqrt{25-x^{2}}$, $-4 \leq x \leq 4$

Set up and evaluate the definite integral for the area of the surface generated by revolving the curve about the y-axis. y=∛x+2

An ornamental light bulb is designed by revolving the graph of y = 1/3x^½ - x^3/2, 0 ≤ x ≤ 1/3 about the x-axis, where x and y are measured in feet. Find the surface area of the bulb and use the result to approximate the amount of glass needed to make the bulb. (Assume that the glass is 0.015 inch thick.)

In this exercise, use the integration capabilities of the graphing utility to verify your results.

$$f(x)=x^4-4 x^2, \quad g(x)=x^2-4$$

Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the x-axis. y=√(x+2), y=x, y=0

Use the shell method to write and evaluate the definite integral that represents the volume of the solid generated by revolving the plane region about the y-axis.

Use the shell method to write and evaluate the definite integral that represents the volume of the solid generated by revolving the plane region about the y-axis. $y=x^{3 / 2}, \ y=8, \ x=0$

Use the integration capabilities of the graphing utility to approximate the volume of the solid generated by revolving the region about the y-axis.

$$y=\frac{2}{1+e^{1 / x}}, \quad y=0, \quad x=1, \quad x=3$$