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.

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

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 $M_x, M_y$, 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 $x=6$

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

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

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

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

The radius r of a sphere is increasing at a rate of 3 inches per minute. (a) Find the rates of change of the volume when r = 9 inches and r = 36 inche s. (b) Explain why the rate of change of the volume of the sphere is not constant even though dr/ dt is constant.

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

$$y=2\arctan x,\quad 0\leq x\leq1$$

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

$$\frac{x^2}{16}+\frac{y^2}{9}=1$$

the $x$-axis (prolate spheroid)