Finding the volume of an ellipse watermelon

Jun 2017
8
0
Adelaide
An ellipse watermelon has a major axis 28cm and minor axis 25cm. The equation of the ellipse is y = 12.5 - 12.5x/145 Find the volume using calculusJust wondering if anyone can help me out on this question! I'm not quite sure how to answer it as I keep getting an abnormally large number.Thanks!
 

skeeter

MHF Helper
Jun 2008
16,217
6,765
North Texas
An ellipse watermelon has a major axis 28cm and minor axis 25cm. The equation of the ellipse is y = 12.5 - 12.5x/145
?

$\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$

assuming the major axis is along the x-axis, $a=14$ and $b=\dfrac{25}{2}$ ...

$\dfrac{x^2}{196}+\dfrac{4y^2}{625} = 1$

$y^2 = \dfrac{625}{4}\left(1-\dfrac{x^2}{196}\right)$

\(\displaystyle V = 2\pi \int_0^{14} \dfrac{625}{4}\left(1-\dfrac{x^2}{196}\right) \, dx\)

\(\displaystyle V = \dfrac{625\pi}{2} \int_0^{14} \left(1-\dfrac{x^2}{196}\right) \, dx = \dfrac{8750\pi}{3}\)
 
Jun 2017
8
0
Adelaide
If the question is asking in terms of the solid of revolution, would the x-axis be 12.5 not 14? Hence changing the dimensions of the integration??
 
Jun 2017
8
0
Adelaide
Screen Shot 2017-07-11 at 12.55.57 pm.png
Screen Shot 2017-07-11 at 12.55.57 pm.png

Does this diagram make sense or does it need to be around the other way?? And I also thought that it was considering the whole ellipse not half? I'm a bit confised sorry...
 

skeeter

MHF Helper
Jun 2008
16,217
6,765
North Texas
I centered the ellipse at the origin. I don't think I've ever seen a watermelon looking like it was a solid of revolution about its minor axis.



The integral set up takes advantage of symmetry w/respect to the y-axis.

$\displaystyle \pi \int_{-14}^{14} [r(x)]^2 \, dx = 2\pi \int_0^{14} [r(x)]^2 \, dx$

note $[r(x)]^2 = y^2$
 

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