Determine which value best approximates the length of the arc represented by the integral $\int_{0}^{\pi / 4} \sqrt{1+\left(\sec ^{2} x\right)^{2}} d x$. (Make your selection on the basis of a sketch of the arc and not by performing any calculations.) (a) -2, (b) 1, (c) $\pi$, (d) 4, (e) 3.

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

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.

Which of the following requires more work? Explain your reasoning. (a) A 60-pound box of books is lifted 3 feet. (b) A 60-pound box of books is held 3 feet in the air for 2 minutes.

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

$$y=x^2-4 x+3, \quad y=3+4 x-x^2$$

Explain why fluid pressure on a surface is calculated using horizontal representative rectangles instead of vertical representative rectangles.

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.

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

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

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

$$y=\sqrt{1-x^3}, \quad y=0, \quad x=0$$

(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. y = √1 - x³, y=0, x=0

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.

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

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis. y=cos 2x, y=0, x=0, x=π/4

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

$$f(x)=x^3-2 x+1, g(x)=-2 x, x=1$$