Determine which value best approximates the arc length represented by the integral. Make your selection on the basis of a sketch of the arc, not by performing calculations.

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

$$\text{value} = 7$$

Sketch the region bounded by the graphs of the equations and find the area of the region.

$$y=-x^3+3, \quad y=x, \quad x=-1, \quad x=1$$

In Exercise given below, find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line y=4.

y=x+4, y=4, x=4

In this exercise, use the shell method to write and evaluate the definite integral that represents the volume of the solid generated by revolving the plane region bounded by the graphs of the equations about the $x$-axis.

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

Find the arc length of the graph of the function over the indicated interval.

$$x=\frac{1}{3} \sqrt{y}(y-3), \quad 1 \leq y \leq 4$$

Write and evaluate the definite integral that represents the volume of the solid formed by revolving the region bounded by the graphs of the equations about the $y$-axis.

$$y=8 /(x-2), \quad y=1, \quad y=8, \quad x=0$$

Sketch the region bounded by the graphs of the equations and find the area of the region.

$$f(x)=-\frac{4}{x^3}, \quad y=0, \quad x=-3 \quad x=-1$$

In this exercise, use the shell method to write and evaluate the definite integral that represents the volume of the solid generated by revolving the plane region bounded by the graphs of the equations about the $x$-axis.

$$y=4 x^2, \quad x=0, \quad y=4$$

Find the arc length of the graph of the function over the indicated interval.

$$x=\frac{1}{3}\left(y^2+2\right)^{3 / 2}, \quad 0 \leq y \leq 4$$

Sketch the region bounded by the graphs of the equations and find the area of the region.

$$f(x)=\frac{1}{9 x^2}, \quad y=1, \quad x=1, \quad x=2$$

Write and evaluate the definite integral that represents the volume of the solid formed by revolving the region bounded by the graphs of the equations about the $y$-axis.

$$y=\sqrt{5 x+1}, \quad y=6, \quad x=0$$

In this exercise, use the shell method to write and evaluate the definite integral that represents the volume of the solid generated by revolving the plane region bounded by the graphs of the equations about the $x$-axis.

$$y=x^3, \quad x=0, \quad y=8$$

Consider the region $T$ bounded by the graphs of $y=x^2$, $y=-2 x$, and $x=2$.

Write, but do not evaluate, an expression involving one or more integrals that gives the perimeter of $T$.

Consider the region $T$ bounded by the graphs of $y=x^2$, $y=-2 x$, and $x=2$.

Find the volume of the solid formed by revolving $T$ about the horizontal line $y=-4$.