In this exercise, determine which value best approximates the length of the arc represented by the integral. (Make your selection on the basis of a sketch of the arc and not by performing any calculations.)

$$\displaystyle\int_0^2\sqrt{1+\left[\frac{d}{d x}\left(\frac{5}{x^2+1}\right)\right]^2} d x$$

$$-4$$

In this exercise, sketch the region bounded by the graphs of the algebraic functions and find the area of the region.

$$f(x)=\sqrt{3x}+1,\quad g(x)=x+1$$

In this exercise, set up and evaluate the integrals for finding the area and moments about the $x$- and $y$-axes for the region bounded by the graphs of the equations. (Assume $p=1$)

$$y=x^2,y=x$$

In this exercise, find the volume of the solid generated by revolving the region bounded by the graphs of the equationsabout the $x$-axis.

$$y=\dfrac{1}{x},\quad y=0,\quad x=1,\quad x=4$$

A porthole on a vertical side of a submarine (submerged in seawater) is $1$ square foot. Find the fluid force on the porthole, assuming that the center of the square is $15$ feet below the surface.

In this exercise, find the volume of the solid generated by revolving the plane region bounded by the equations about the indicated line(s).

$y=\frac{1}{x^4+1}, y=0, x=0, x=1$ revolved about the $y$-axis

In this exercise, use the integration capabilities of a graphing utility to approximate the arc length.

$$x=\sqrt{36-y^2},\quad0\leq y\leq3$$

In this exercise, find the volume of the solid generated by revolving the plane region bounded by the equations about the indicated line(s).

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$

the $x$-axis (prolate spheroid)

In this exercise, find 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

$$x=\sqrt{36-y^2},\quad0\leq y\leq3$$

In this exercise, find the volume of the solid generated by revolving the plane region bounded by the equations about the indicated line(s).

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$

the y-axis (oblate spheroid)

In this exercise, graph the function, highlighting the part indicated by the given interval.

$$x=\sqrt{36-y^2},\quad0\leq y\leq3$$

In this exercise, find the volume of the solid generated by revolving the region bounded by the graphs of the equationsabout the $x$-axis.

$$y=x \sqrt{4-x^2}, \quad y=0$$

In this exercise, find $M_x, M_y$, and $(\bar{x}, \bar{y})$ for the laminas of uniform density $\rho$ bounded by the graphs of the equations.

$$x=y+2,x=y^2$$

In this exercise, use the shell method to find the volume of the solid generated by revolving the plane region about the given line.

$y=\sqrt{x},\quad y=0,\quad x=4$, about the line $x=6$