Exercises - Volumes of Revolution

  1. Find the volumes of the solids generated wehen the region bounded by $y=\sqrt{x}; \quad y=0;$ and $x=4$ is revolved about: the $x$-axis; the $y$-axis; the line $y=5$; $y=-1$; $x=6$; and $x=-3$.

  2. Setup (but do not evaluate) the integral(s) necessary to calculate the volume of the solid generated when the region bounded by $y=2\sqrt{x}; \ y=1; \ y=2; \ \textrm{and} \ x=4$ is revolved about $y=5$. Do the same for the region revolved about $x=-1$

  3. Setup (but do not evaluate) the integral(s) necessary to calculate the volume of the solid generated when the region bounded by $x^2+y^2=36; \ \textrm{and} \ y=0$ is revolved about $y=-3$. Do the same for the region revolved about $x=6$.

  4. Setup (but do not evaluate) the integral(s) necessary to calculate the volume of the solid generated when the region bounded by $y=\sin x; \ y=0; \ x=0; \ \textrm{and} \ x=\pi$ is revolved about the $x$-axis

  5. Find the volume f the solids generated by revolving the region bounded by $x=y^2+1$, $y=2$, $x=0$, and $y=0$ about the following: the $x$-axis; the line $y=4$; $y=-1$; $x=7$; and $x=-3$.

  6. Setup (but do not evaluate) the integral(s) necessary to calculate the volume of the solid generated when the region bounded by $y=\sqrt{4-x^2}; \ \textrm{and} \ y=1$ is revolved about: the $x$-axis; $y=1/2$; $y=2$; $x=2$; and $x=-5/2$.

  7. Setup (but do not evaluate) the integral(s) necessary to calculate the volume of the solid generated when the region bounded by $y=\cos x; \ y=0; \ x=0; \ \textrm{and} \ x=\frac{\pi}{2}$ is revolved about: the $y$-axis.

  8. Sketch the region bounded by $y=\sqrt{4-x^2}; \ y = x-2; \textrm{ and } x=0$. Find, using both the shell and disk methods, the volumes of the solid generated by revolving this region about the $y$-axis. Do the same for the solid generated by revolving the region about the $x$-axis.

  9. Find, using both the shell and disk methods, the volume generated when the region bounded by $y=x^2-1; \ y=2x+2; \ x=0; \ \textrm{ and } y=0$ is revolved about $x=-2$. Do the same when the region is revolved about $y=10$.

  10. Sketch the region enclosed by $x=3y$ and $x=-y^2+4$. Give the integral(s) that represent the volume generated when this region is revolved about $y=4$, using both the shell and the disk methods.