Rewrite the equations of the given boundary lines:
<em>y</em> = -<em>x</em> + 1 ==> <em>x</em> + <em>y</em> = 1
<em>y</em> = -<em>x</em> + 4 ==> <em>x</em> + <em>y</em> = 4
<em>y</em> = 2<em>x</em> + 2 ==> -2<em>x</em> + <em>y</em> = 2
<em>y</em> = 2<em>x</em> + 5 ==> -2<em>x</em> + <em>y</em> = 5
This tells us the parallelogram in the <em>x</em>-<em>y</em> plane corresponds to the rectangle in the <em>u</em>-<em>v</em> plane with 1 ≤ <em>u</em> ≤ 4 and 2 ≤ <em>v</em> ≤ 5.
Compute the Jacobian determinant for this change of coordinates:
![J=\begin{bmatrix}\frac{\partial u}{\partial x}&\frac{\partial u}{\partial y}\\\frac{\partial v}{\partial x}&\frac{\partial v}{\partial y}\end{bmatrix}=\begin{bmatrix}1&1\\-2&1\end{bmatrix}\implies|\det J|=3](https://tex.z-dn.net/?f=J%3D%5Cbegin%7Bbmatrix%7D%5Cfrac%7B%5Cpartial%20u%7D%7B%5Cpartial%20x%7D%26%5Cfrac%7B%5Cpartial%20u%7D%7B%5Cpartial%20y%7D%5C%5C%5Cfrac%7B%5Cpartial%20v%7D%7B%5Cpartial%20x%7D%26%5Cfrac%7B%5Cpartial%20v%7D%7B%5Cpartial%20y%7D%5Cend%7Bbmatrix%7D%3D%5Cbegin%7Bbmatrix%7D1%261%5C%5C-2%261%5Cend%7Bbmatrix%7D%5Cimplies%7C%5Cdet%20J%7C%3D3)
Rewrite the integrand:
![-3x+4y=-3\cdot\dfrac{u-v}3+4\cdot\dfrac{2u+v}3=\dfrac{5u+7v}3](https://tex.z-dn.net/?f=-3x%2B4y%3D-3%5Ccdot%5Cdfrac%7Bu-v%7D3%2B4%5Ccdot%5Cdfrac%7B2u%2Bv%7D3%3D%5Cdfrac%7B5u%2B7v%7D3)
The integral is then
![\displaystyle\iint_R(-3x+4y)\,\mathrm dx\,\mathrm dy=3\iint_{R'}\frac{5u+7v}3\,\mathrm du\,\mathrm dv=\int_2^5\int_1^45u+7v\,\mathrm du\,\mathrm dv=\boxed{333}](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Ciint_R%28-3x%2B4y%29%5C%2C%5Cmathrm%20dx%5C%2C%5Cmathrm%20dy%3D3%5Ciint_%7BR%27%7D%5Cfrac%7B5u%2B7v%7D3%5C%2C%5Cmathrm%20du%5C%2C%5Cmathrm%20dv%3D%5Cint_2%5E5%5Cint_1%5E45u%2B7v%5C%2C%5Cmathrm%20du%5C%2C%5Cmathrm%20dv%3D%5Cboxed%7B333%7D)
Answers:
- sin(C) = 24/25
- cos(Z) = 5/13
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Explanation:
The formulas we use are
- sin(angle) = opposite/hypotenuse
- cos(angle) = adjacent/hypotenuse
The fraction 15/39 reduces to 5/13 after dividing both parts by the GCF 3.
Answer:
0
Step-by-step explanation:
Imagine that you have zero cookies and you split them evenly among zero friends. How many cookies does each person get? See? It doesn't make sense. And Cookie Monster is sad that there are no cookies, and you are sad that you have no friends.
It’s the first one. I think
Answer:
y = 2/3x +2
Step-by-step explanation:
First we need to find the slope
m = (y2-y1)/(x2-x1)
= (6-2)/(6-0)
=4/6
=2/3
The slope intercept form of the equation for a line is
y= mx+b
y =2/3 x +b
We know the y intercept is 2 (the y intercept is when x=0)
y = 2/3x +2