a. Express the area A of the cross-section cut from the

Chapter 11, Problem 45E

(choose chapter or problem)

a. Express the area A of the cross-section cut from the ellipsoid

$x^{2}+\frac{y^{2}}{4}+\frac{z^{2}}{9}=1$

by the plane $z = c$ as a function of $c$. (The area of an ellipse with semiaxes $a$ and $b$ is $\pi a b)$

b. Use slices perpendicular to the $z-\text { axis }$ to find the volume of the ellipsoid in part (a).

c. Now find the volume of the ellipsoid

$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1$

Does your formula give the volume of a sphere of radius $a$ if $a = b = c$ ?

Equation Transcription:

${x}^{2}\ +\ \frac{{y}^{2}}{4}\ +\ \frac{{z}^{2}}{9}\ =\ 1$

$z=c$

$c$

$a$

$b$

$\pi{}ab$

$z-axis$

$\frac{{x}^{2}}{{a}^{2}}\ +\ \frac{{y}^{2}}{{b}^{2}}\ +\ \frac{{z}^{2}}{{c}^{2}}\ =\ 1.$

$a$

$a=b=c$

Text Transcription:

x^2 + y^2/4 + z^2/9 = 1

z=c

Pi ab

z-axis

x^2/a^2 + y^2/b^2 + z^2/c^2 = 1.

a=b=c

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