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

$$y=\sqrt[3]{(x-2)^2(x-6)^2}, \quad y=0, \quad x=2, \quad x=6$$

Write and evaluate the definite integral that represents the area of the surface generated by revolving the curve on the indicated interval about the y-axis. $y=\sqrt[3]{x}$, $1 \leq x \leq 2$

Find the arc length from (0, 3) clockwise to (2, √5) along the circle x²+y²=9.

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 y-axis. y = √x

(a) use a graphing utility to graph the region bounded by the graphs of the equations, (b) find the area of the region analytically, and (c) use the integration capabilities of the graphing utility to verify your results. f(x) = 1 / (1 + x²), g(x) = ½x²

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis. Verify your results using the integration capabilities of a graphing utility. y = e^(x-1), y=0, x=1, x=2

Find the area of the region. $f(x)=x\left(x^{2}-3 x+3\right), \ g(x)=x^{2}$

A force of 5 pounds is needed to stretch a spring 1 inch from its natural position. Find the work done in stretching the spring from its natural length of 10 inches to a length of 15 inches.

A cable of a suspension bridge forms a catenary modeled by the equation $y=300 \cosh \left(\frac{x}{2000}\right)-280, \ -2000 \leq x \leq 2000$ where x and y are measured in feet. Use a graphing utility to approximate the length of the cable.

Find the center of mass of the lamina in previous exercise when the circular portion of the lamina has twice the density of the square portion of the lamina.

In this exercise, 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.

$$y=3 x, \quad 0 \leq x \leq 3$$

Use a graphing utility to graph the region bounded by the graphs of the equations. $f(x)=x\left(x^{2}-3 x+3\right), \ g(x)=x^{2}$

Does it take any work to push an object that does not move? Explain.

Neglecting air resistance and the weight of the propellant, determine the work done in propelling a five-metric-ton satellite to a height of 200 miles above Earth.