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Find the volume of the solid bounded above by the surface $z=f(x, y)$ and below by the region $R$ in the $x$ -y plane. $f(x, y)=12-\frac{3}{2} x-2 y, R: 3 x+4 y=24, x=0$ and $y=0$.

$$96$$

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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01:26

Find the volume of the sol…

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04:30

03:14

03:44

for this problem we are asked to find the volume of the solid bounded above by the surface. Z equals 12 minus 3/2. X minus two. Y and below by the region defined by three X plus four, Y equals 24 X equals zero and y equals zero in the xy plane. So to begin we want to plot out our projection of r in the xy plane. Now we have a few different ways that we can set this up this red line here. Is that line three X plus four? Y equals 24. So what I'll do just for the sake of habit, I'll treat this as a horizontally simple region. Which means that we are bounded below by y equals zero And bounded above by four. Y equals 24 -3 x. Or why equals 24 -3 or 24 -3 X. All over four. We could then rewrite as y equals 6 -3, x over four. So this will then be the integral from now integrating X. We can see that we'd be integrating from 0-2. And then why is between zero and 6 -3? X over four and ar inte Grant is 12 minus 3/2. X minus two. Y dy dx. So first integrating over why we'll have that. Our innermost integral Will become 12 Y -3/2. X y minus two. Y squared over two or minus Y squared Evaluated from 0 to 6 -3 x over four dx. We can then simplify or evaluate out and simplify that into Grant. To write it as 36 minus nine X Plus nine x squared over 16 dx. She will then become 36 x minus nine X squared over two Plus nine x cubed over 16 times three Integrated from 0-8, which will give us the final result 96.

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