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YZ

# \begin{array}{c}{\text { Volume of an ellipsoid Find the volume of the ellipsoid }} \\ {\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1} \\ {\text { (Hint: Let } x=a u, y=b v, \text { and } z=c w . \text { Then find the volume of }} \\ {\text { an appropriate region in } u v w-\text { space.) }}\end{array}

## $\frac{4 \pi}{3} \times(a b c)^{3}$

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### Video Transcript

So first we're gonna establish the Chico Been with respect over U B and W jiko Been is given by that of x y z over that you be w the relation are executes a you Why? Who's PB and Z u c w So the judge cobian will be ABC on the diagonal and other parts are just Cyril. That's over the evidence. So it's a diving matrix. We don't know countries ABC. So the are determinants is ABC Okay? Now on the U V w plane this the tips toys will be transformed into a unit bull with the equation you swore close piecework last ever scored less than equal to one his integral echoes ABC, which is a tree Coben times to you the V w over the region G which is transformed and which is unit ball. So the answer will be applied the, uh, geometrical boarding from therefore also that's gonna be 3.4 hour three and the hearing him ABC here, so Yeah, but no answer will be full power three times, ABC

YZ
University of Illinois at Urbana-Champaign

#### Topics

Limits

Integrals

Multiple Integrals

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