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

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

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

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

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)

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 $y$-axis (oblate spheroid)

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

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

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

$y=4x-x^2,\quad y=0$, about the line $x=5$

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.

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

Fluid Force on a Rectangular Plate A rectangular plate of height $h$ feet and base $b$ feet is submerged vertically in a tank of fluid that weighs $w$ pounds per cubic foot. The center is $k$ feet below the surface of the fluid, where $h \leq k / 2$. Show that the fluid force on the surface of the plate is

$$F=w k h b .$$

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

$$y=x,\quad y=2-x,\quad y=0$$

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

$$y=\dfrac{1}{\sqrt{x+1}}, \quad y=0, \quad x=0, \quad x=3$$

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

$$y=\sqrt{x}, y=2, x=0$$

the line $x=-1$

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

$$y=\ln x,\quad 1\leq x\leq5$$

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

$$y=\sqrt{x}, y=2, x=0$$

the $y$-axis