Answer to Question #106079 in Calculus for kawthar

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 }\$ $=500\pi -\frac { 500\pi }{ 3 } =\frac { 1000\pi }{ 3 }$