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 shell method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the $y$-axis.

$$y=\frac{1}{x^2}, \quad y=0, \quad x=2, \quad x=5$$

Use the integration capabilities of a graphing utility to approximate the arc length.

$$y=2 \arctan x, \quad[0,1]$$

Use the integration capabilities of a graphing utility to approximate the arc length.

$$y=2 \arctan x, \quad[0,1]$$

(a) Sketch the graph of the function, highlighting the part indicated by the given interval, (b) write a definite integral that represents the arc length of the curve over the indicated interval and observe that the integral cannot be evaluated with the techniques studied so far, and (c) use the integration capabilities of a graphing utility to approximate the arc length. y=ln x, [1, 5]

(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)=x(x²-3x+3), g(x)=x²

(a) Sketch the graph of the function, highlighting the part indicated by the given interval, (b) write a definite integral that represents the arc length of the curve over the indicated interval and observe that the integral cannot be evaluated with the techniques studied so far, and (c) use the integration capabilities of a graphing utility to approximate the arc length. $y=2 \arctan x$, [0, 1]

Write a definite integral that represents the arc length of the curve over the indicated interval and observe that the integral cannot be evaluated with the techniques studied so far

$$y=2 \arctan x, \quad[0,1]$$

Write a definite integral that represents the arc length of the curve over the indicated interval and observe that the integral cannot be evaluated with the techniques studied so far.

$$y=2 \arctan x, \quad[0,1]$$

Sketch the graph of the function, highlighting the part indicated by the given interval.

$$y=2 \arctan x, \quad[0,1]$$

Sketch the graph of the function, highlighting the part indicated by the given interval.

$$y=2 \arctan x, \quad[0,1]$$

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.

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis. $y=\frac{6}{x}$, y=0, x=1, x=3

Fnd the volume of the solid generated by revolving the region bounded by the graphs of the equations about the $x$-axis.

$$y=\frac{6}{x}, \quad y=0, \quad x=1, \quad x=3$$