Determining the Volume of a Solid of Revolution

Key Questions

  • How do you find the volume of a cone using an integral?

    A cone with base radius r and height h can be obtained by rotating the region under the line y=r/hx about the x-axis from x=0 to x=h.
    By Disk Method,
    V=pi int_0^h(r/hx)^2 dx={pi r^2}/{h^2}int_0^hx^2 dx
    by Power Rule,
    ={pir^2}/h^2[x^3/3]_0^h={pir^2}/{h^2}cdot h^3/3=1/3pir^2h

    Wataru · · Sep 8 2014
  • How do you use an integral to find the volume of a solid torus?

    If the radius of its circular cross section is r, and the radius of the circle traced by the center of the cross sections is R, then the volume of the torus is V=2pi^2r^2R.

    Let's say the torus is obtained by rotating the circular region x^2+(y-R)^2=r^2 about the x-axis. Notice that this circular region is the region between the curves: y=sqrt{r^2-x^2}+R and y=-sqrt{r^2-x^2}+R.

    By Washer Method, the volume of the solid of revolution can be expressed as:
    V=pi int_{-r}^r[(sqrt{r^2-x^2}+R)^2-(-sqrt{r^2-x^2}+R)^2]dx,
    which simplifies to:
    V=4piR\int_{-r}^r sqrt{r^2-x^2}dx
    Since the integral above is equivalent to the area of a semicircle with radius r, we have
    V=4piRcdot1/2pi r^2=2pi^2r^2R

    Wataru · 4 · Aug 30 2014

Questions

Applications of Definite Integrals