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Homework Statement:
 Find the volume of a circular cone.
Relevant Equations:
 N/A
This is not homework. I have given myself two parameters; ##\theta##, and ##\alpha##. (see figure, it is a side view):
The idea is to find an expression for the radius of the circles as ##x## varies on that line (figure), then sum up infinitely many cylinders of infinitesimal thickness.
The radii all have an angle ##\theta## with the ##x##axis, and meet with the line making an angle ##\alpha## with the same axis at ##x=b##, thus:
$$r(x)=\sqrt{(bx)^2+(b\tan\alpha)^2}$$
To find ##b##, I need to solve ##(\tan\alpha)x\tan\theta(xb)=0##, which gives me:
$$b=\frac{\tan\theta\tan\alpha}{\tan\theta}x$$
If the line upon which I put my ##x##axis has a length ##a##, then the volume of the cone is:
$$\pi\int_0^ar^2(x)dx$$
Is the reasoning correct?
EDIT:
I could've also remembered triangles' similarities and notice that ##\frac{R}{a}=\frac{r(x)}{x}##.
The idea is to find an expression for the radius of the circles as ##x## varies on that line (figure), then sum up infinitely many cylinders of infinitesimal thickness.
The radii all have an angle ##\theta## with the ##x##axis, and meet with the line making an angle ##\alpha## with the same axis at ##x=b##, thus:
$$r(x)=\sqrt{(bx)^2+(b\tan\alpha)^2}$$
To find ##b##, I need to solve ##(\tan\alpha)x\tan\theta(xb)=0##, which gives me:
$$b=\frac{\tan\theta\tan\alpha}{\tan\theta}x$$
If the line upon which I put my ##x##axis has a length ##a##, then the volume of the cone is:
$$\pi\int_0^ar^2(x)dx$$
Is the reasoning correct?
EDIT:
I could've also remembered triangles' similarities and notice that ##\frac{R}{a}=\frac{r(x)}{x}##.
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