Determining the Volume of a Solid of Revolution
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Key Questions

A cone with base radius
#r# and height#h# can be obtained by rotating the region under the line#y=r/hx# about the xaxis 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# 
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+(yR)^2=r^2# about the#x# axis. Notice that this circular region is the region between the curves:#y=sqrt{r^2x^2}+R# and#y=sqrt{r^2x^2}+R# .By Washer Method, the volume of the solid of revolution can be expressed as:
#V=pi int_{r}^r[(sqrt{r^2x^2}+R)^2(sqrt{r^2x^2}+R)^2]dx# ,
which simplifies to:
#V=4piR\int_{r}^r sqrt{r^2x^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# 
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Applications of Definite Integrals

1Solving Separable Differential Equations

2Slope Fields

3Exponential Growth and Decay Models

4Logistic Growth Models

5Net Change: Motion on a Line

6Determining the Surface Area of a Solid of Revolution

7Determining the Length of a Curve

8Determining the Volume of a Solid of Revolution

9Determining Work and Fluid Force

10The Average Value of a Function