Revolution about its axis question

[yohanson77]

Hi all,

Well I've moved on slightly and now I'm having a go at calculus. I have this question and it's playing on my mind:

Q. A cement silo is constructed in the form of a frustrum of a cone, having a base diameter of 2.6m, upper diameter of 4.8m and a vertical height of 6.7m. Considering this is a volume of revolution about its axis of symmetry, as shown below, use calculus to determine the capacity of the silo when full?

I've worked on a sphere one and found the radius ok, this has me over a barrel.

Please help.

yours

yohanson :?

 

[skeeter]

plot the following two points on an set of xy axes ...

(0, 1.3), (6.7, 2.4)

connect the two points with a line segment ... rotating this segment about the x-axis will form the described solid (laying on its side)

the equation of the segment joining the two points is y = (11/67)x + 1.3

the volume of the solid can be defined by the definite integral ...

\(\displaystyle \L V = \pi \int_0^{6.7} \left(\frac{11}{67}x + 1.3\right)^2 \, dx = \frac{70819 \pi}{3000}\)

edit ... yup, forgot to square the integrand

 

[pka]

\(\displaystyle \L V = \pi \int_0^{6.7} \frac{11}{67}x + 1.3 \, dx = \frac{2479 \pi}{200}\)

\(\displaystyle \L V = \pi \int_0^{6.7} (\frac{11}{67}x + 1.3)^2 \, dx\)????

 

[galactus]

Check against the formula for the volume of a cone frustum.

\(\displaystyle \L\\\frac{1}{3}{\pi}h(r^{2}+rR+R^{2})\)