Determining the Volume of a Solid of Revolution
Key Questions
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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# -
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#
Questions
Applications of Definite Integrals
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Solving Separable Differential Equations
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Slope Fields
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Exponential Growth and Decay Models
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Logistic Growth Models
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Net Change: Motion on a Line
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Determining the Surface Area of a Solid of Revolution
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Determining the Length of a Curve
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Determining the Volume of a Solid of Revolution
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Determining Work and Fluid Force
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The Average Value of a Function