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Evaluate the double integral $\int_{R} \int f(x, y) d A$ over the indicated region $R$, for the given function $f$. $f(x, y)=6 x^{2} y, R$ is the region bounded by $y=2 x^{2}$ and $y=2 \sqrt{x}$.

$$9 / 7$$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 6

Double Integrals

Partial Derivatives

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12:15

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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Evaluate the double integr…

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for this problem we are asked to evaluate the double integral of six X squared Y over the region bounded by Y equals two X squared and y equals to brood X. So our first step here is to plot out our region so we can see that we're going to have something that's a little bit complicated no matter how we try to set this up, but I'm going to treat this as a horizontally simple region. So our integral then will be the integral Over X. from 0 to 1 of the integral over why from bounded below would be two X squared. And the upper bound would be to root X. And we're integrating six X squared Y dy dx. So integrating six X squared Y over Y. First gives us six X squared Y squared over two. Or just three X squared Y squared evaluated from two X squared up to two root X D X. So we'll find then that if we evaluate from end to end, that inner term, we have that this becomes the integral from 0 to 1 of 12 X cubed minus 12 X. To the power of six dx. Which will then become 12 X power 4/4. 12/4 is three. So we have three X. The power four minus 12 X. The power of 7/7. So just minus 12/7 X power of seven evaluated from 0 to 1. Which gives us our final result then of 9/7

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