Question

Find the volume of the ellipsoid generated by revolving the ellipse
with the lengths of Major and Minor axis respectively as 20cm and
10cm about its major axis.

Answer 1

ANSWER $\frac { 1000\pi }{ 3 } \quad$

EXPLANATION. The volume of the body formed by the rotation of the curve $G=\left\{ \left( x,f(x) \right) :a\le x\le b \right\}$ around the x-axis is calculated by the integral $\pi \int _{ a }^{ b }{ { f }^{ 2 } } (x)dx$ . Ellipse with the axis 20cm and 10cm defined by the equation $\frac { { x }^{ 2 } }{ { 10 }^{ 2 } } +\frac { { y }^{ 2 } }{ { 5 }^{ 2 } } =1$ .Therefore the ellipsoid is formed by the rotation of the curve $G=\left\{ (x,f(x)):{ f }^{ 2 }(x)=25\left( 1-\frac { { x }^{ 2 } }{ { 10 }^{ 2 } } \right) ,-10\le x\le 10 \right\}$ . The volume of the ellipsoid is $\pi \int _{ -10 }^{ 10 }{ 25\left( 1-\frac { { x }^{ 2 } }{ { 10 }^{ 2 } } \right) dx= } 2\pi \ \int _{ 0 }^{ 10 }{ \left( 25-\frac { { x }^{ 2 } }{ 4 } \right) } dx=$

$= 500π -$ \frac { \pi }{ 2 } { ( \frac { { x }^{ 3 } }{ 3 } \ ) | }_{ 0 }^{ 10 }\ $=500\pi -\frac { 500\pi }{ 3 } =\frac { 1000\pi }{ 3 }$