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YZ

# Let $D$ be the region in $x y z$ -space defined by the inequalities$$1 \leq x \leq 2, \quad 0 \leq x y \leq 2, \quad 0 \leq z \leq 1$$Evaluate$$\iiint_{D}\left(x^{2} y+3 x y z\right) d x d y d z$$by applying the transformation$$u=x, \quad v=x y, \quad w=3 z$$and integrating over an appropriate region $G$ in $u v w$ -space.

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##### Lily A.

Johns Hopkins University

##### Heather Z.

Oregon State University

##### Kristen K.

University of Michigan - Ann Arbor

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### Video Transcript

So now we have the relationship. You goes to x vehicles too. That's why in the biggest three z waste. Absolutely. Kobe. So this implies actually, on X equals you. Why? Cause to be over you and Zeke comes to Debbie over three. Jacoby is pretty simple. We had 100 Next time you be over your squared whenever you 000 1/3 so calculate this determinants. We're gonna have one word for you. Hence the integral will be transformed into over you be that plane that's called this region tree and the Inter Gran's equals two. You swear I'm sick. Be over you plus three you times be over you comes 1/3 of w comes to drink Oh, being witches One word for you I'm sti u d v d w We can't do it This and the answer would be on two times three two plus three times long. Listen, Misha. Level two

YZ
University of Illinois at Urbana-Champaign

#### Topics

Limits

Integrals

Multiple Integrals

##### Lily A.

Johns Hopkins University

##### Heather Z.

Oregon State University

##### Kristen K.

University of Michigan - Ann Arbor

Lectures

Join Bootcamp