In this exercise, 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.

$$y=\cos x, \quad y=0, \quad x=0, \quad x=\dfrac{\pi}{2}$$

Two identical semicircular windows are placed at the same depth in the vertical wall of an aquarium. Which is subjected to the greater fluid force? Explain.

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

$x^{4 / 3}+y^{4 / 3}=1, \quad x=0, \quad y=0$, first quadrant

(a) use a graphing utility to graph the plane region bounded by the graphs of the equations, and (b) use the integration capabilities of the graphing utility to approximate the volume of the solid generated by revolving the region about the y-axis. x^4/3 + y^4/3 = 1, x = 0, y = 0, first quadrant.

In this exercise, find and/or verify the centroid of the common region used in engineering.

Triangle Show that the centroid of the triangle with vertices $(-a, 0),(a, 0)$, and $(b, c)$ is the point of intersection of the medians (see figure).In this exercise, find and/or verify the centroid of the common region used in engineering.

Triangle Show that the centroid of the triangle with vertices $(-a, 0),(a, 0)$, and $(b, c)$ is the point of intersection of the medians (see figure).

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis. y = sin x, y = 0, x = 0, x = π

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

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

$$f(x)=x\left(x^2-3 x+3\right), g(x)=x^2$$

Consider a 20-foot chain that weighs 3 pounds per foot hanging from a winch 20 feet above ground level. Find the work done by the winch in winding up the specified amount of chain. -Run the winch until the bottom of the chain is at the 10-foot level.

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

Find the arc length of the graph of the function over the given interval. $f(x)=\frac{4}{5} x^{5 / 4}, \ [0,4]$

Electrical wires suspended between two towers form a catenary modeled by the equation y = 20 cosh x/20, -20 ≤ x ≤ 20 where x and y are measured in meters. The towers are 40 meters apart. Find the length of the suspended cable.

The base of a solid is a circle of radius a, and its vertical cross sections are equilateral triangles. The volume of the solid is 10 cubic meters. Find the radius of the circle.

consider a 20-foot chain that weighs 3 pounds per foot hanging from a winch 20 feet above ground level. Find the work done by the winch in winding up the specified amount of chain. -Wind up one-third of the chain.