asked 2014-07-30 10:08:58 -0600
So I am completely confused on how to solve #1 on the final exam review sheet:
Let $D$ denote the solid pyramid with vertices located at $(4,0,0),(0,1,0),(0,0,2)$ and the origin. Set up $$\iiint_D f(x,y,z)dV$$ in the order $dzdydx$.
We did one similar to this in class but we were given an equation for the plane. I guess the real question is, how would we go about finding an equation of a plane that we could work with?
answered 2014-07-30 15:41:14 -0600
I agree with Wes about almost everything that he did. However, when I compute the cross product, I obtain the vector $\langle 2, 8, 4 \rangle$ (notice that the $z$ component is equal to $4$, not $5$. This results in the following equation for the plane:
$$2(x-4) + 8(y-0) + 4(z-0) = 0$$ $$2x - 8 + 8y + 4z = 0$$ $$2x + 8y + 4z = 8$$
We can verify that this is the correct equation by plugging in all three of our original points to make sure they satisfy this equation.
$(4, 0, 0)$: $$2(4) + 8(0) + 4(0) = 8$$ $$ 8 = 8 $$ The point $(4, 0, 0)$ works and is legit.
$(0, 1, 0)$: $$2(0) + 8(1) + 4(0) = 8$$ $$ 8 = 8 $$ The point $(0, 1, 0)$ works and is legit.
$(0, 0, 2)$: $$2(0) + 8(0) + 4(2) = 8$$ $$ 8 = 8 $$ The point $(0, 0, 2)$ works and is legit.
All three points work, so $2x + 8y + 4z = 8$ is an equation for the plane in question.
answered 2014-07-30 10:50:02 -0600
We find the equation of a plane by taking the cross product of 2 vectors from the differences between 2 sets of points. So we start with our points. Let's make $A=(4,0,0)$, $B = (0,1,0)$, and $C=(0,0,2)$. If we take the differences between the points $A$ and $B$, and $B$ and $C$, we create vectors $\overrightarrow{AB} = \langle 4, -1, 0 \rangle$ and $\overrightarrow{BC} = \langle 0,1,-2\rangle$. We then take the cross product of these vectors $\overrightarrow{AB} \times \overrightarrow{BC}$ and get the matrix $$\begin{vmatrix} I & J & K\ 4 & -1 & 0\ 0 & 1& -2\ \end{vmatrix} $$ This matrix equals $2I -8J +5K$, and if we pick our favorite of the three original points, we can create an equation. I chose point $A$ and found the equation $$2(x -4) -8(y-0)+5(z-0) = 0$$ which, if we distribute, equals $$2x-8y+5z = 8$$ which we can easily use to find our integral. I hope this helps!
EDIT: Well, I don't know how to get the matrix working, but pretend it's 3 rows and three columns!
answered 2014-07-30 20:31:06 -0600
So with using the equation that Justin came up with for the plane, do we just use that equation as our $ f(x,y,z) $ ?If so, I ended up with $$ \int _0 ^4 \int _0^1 \int _0^2 2x +8y +4z \delta z \delta y \delta x $$
and I'm not sure if that was the right way to go from here?
Asked: 2014-07-30 10:08:58 -0600
Seen: 29 times
Last updated: Jul 30 '14
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