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(a) If $f(x, y)=k x^{2} y^{2}$ is a joint $p d f$ over the region $R$ in Exercise $12,$ determine $k$. (b) Determine $\operatorname{Pr}(E)$ if $E$ is the subset of $R$ bounded by $y=6 x, y=x^{2},$ from $x=0$ to $x=1$.

(a) $1 / 186624$(b) $323 / 60279552$

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Frank L.

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 6

Double Integrals

Partial Derivatives

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Oregon State University

Harvey Mudd College

University of Nottingham

Lectures

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In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

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Use Green's Theorem t…

So what we want to do, um, is used Green Sturm. So first we're going to look at the partial derivative of X squared with respect to X. So it's obviously going to give us two X and then the partial derivative of zero with respect to why is obviously going to be zero. So as a result, what will end up getting is, um, we'll get that it's one over to a Yeah, I'm being a rule of X squared ey. And that is the X coordinate of and what we're looking for. And then for the white coordinate we see that will get okay, a negative one over to a Yeah, empty, Integral. If we look at zero, uh, partial derivative of zero with respect to X zero sum will end up getting is minus the y squared or in this case, uh, plus y squared and the derivative of that. So we'll end up getting Why squared the a za result and that's what we expect to get. So based on that, we see that our results have been verified and all you have to do is take the partial derivative of the values to prove it

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