The value of the integral 3ydx+2xdy using Green's theorem be - xy
The value of
be -xy
<h3>What is Green's theorem?</h3>
Green's theorem relates a line integral around a simple closed curve C to a double integral over the plane region D bounded by C.
If M and N are functions of (x, y) defined on an open region containing D and having continuous partial derivatives there, then
= ![\int\int\〖(N_{x}-M_{y}) \;dxdy](https://tex.z-dn.net/?f=%5Cint%5Cint%5C%E3%80%96%28N_%7Bx%7D-M_%7By%7D%29%20%5C%3Bdxdy)
Using green's theorem, we have
=
............................... (1)
Here
= differentiation of function N w.r.t. x
= differentiation of function M w.r.t. y
Given function is: 3ydx + 2xdy
On comparing with equation (1), we get
M = 3y, N = 2x
Now,
= ![\Luge\frac{dN}{dx}](https://tex.z-dn.net/?f=%5CLuge%5Cfrac%7BdN%7D%7Bdx%7D)
= ![\frac{d}{dx} (2x)](https://tex.z-dn.net/?f=%5Cfrac%7Bd%7D%7Bdx%7D%20%282x%29)
= 2
and,
= ![\Huge\frac{dM}{dy}](https://tex.z-dn.net/?f=%5CHuge%5Cfrac%7BdM%7D%7Bdy%7D)
= ![\frac{d}{dy} (3y)](https://tex.z-dn.net/?f=%5Cfrac%7Bd%7D%7Bdy%7D%20%283y%29)
= 3
Now using Green's theorem
= ![\int\int\〖(2 -3) dx dy](https://tex.z-dn.net/?f=%5Cint%5Cint%5C%E3%80%96%282%20-3%29%20dx%20dy)
= ![\int\int\ -dxdy](https://tex.z-dn.net/?f=%5Cint%5Cint%5C%20-dxdy)
= ![-\int\ x dy](https://tex.z-dn.net/?f=-%5Cint%5C%20x%20dy)
=![-xy](https://tex.z-dn.net/?f=-xy)
The value of
be -xy.
Learn more about Green's theorem here:
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Answer:
A: 2,090 glasses of water each day
B: 14,630 gasses of water every every week
C: 438,900 glasses of water every month
D: it would tale about 70 days to drink 1,000,000 glasses of water
Answer:
![\dfrac{5a^3bc^2}{65a^4bc}](https://tex.z-dn.net/?f=%5Cdfrac%7B5a%5E3bc%5E2%7D%7B65a%5E4bc%7D)
Step-by-step explanation:
Of the given choices A, D, E have the correct constant. D has an exponent of <em>a</em> that is too large. E has <em>a</em> in the numerator, not the denominator. So, the only viable choice is the first answer selection.
Answer:
42
Step-by-step explanation: