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LenKa [72]
3 years ago
6

WILL MARK BRAINLIEST!!!! PLEASE HELP

Mathematics
1 answer:
TEA [102]3 years ago
4 0
I’m pretty sure it’s b - standard
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Show that the line integral is independent of path by finding a function f such that ?f = f. c 2xe?ydx (2y ? x2e?ydy, c is any p
Juli2301 [7.4K]
I'm reading this as

\displaystyle\int_C2xe^{-y}\,\mathrm dx+(2y-x^2e^{-y})\,\mathrm dy

with \nabla f=(2xe^{-y},2y-x^2e^{-y}).

The value of the integral will be independent of the path if we can find a function f(x,y) that satisfies the gradient equation above.

You have

\begin{cases}\dfrac{\partial f}{\partial x}=2xe^{-y}\\\\\dfrac{\partial f}{\partial y}=2y-x^2e^{-y}\end{cases}

Integrate \dfrac{\partial f}{\partial x} with respect to x. You get

\displaystyle\int\dfrac{\partial f}{\partial x}\,\mathrm dx=\int2xe^{-y}\,\mathrm dx
f=x^2e^{-y}+g(y)

Differentiate with respect to y. You get

\dfrac{\partial f}{\partial y}=\dfrac{\partial}{\partial y}[x^2e^{-y}+g(y)]
2y-x^2e^{-y}=-x^2e^{-y}+g'(y)
2y=g'(y)

Integrate both sides with respect to y to arrive at

\displaystyle\int2y\,\mathrm dy=\int g'(y)\,\mathrm dy
y^2=g(y)+C
g(y)=y^2+C

So you have

f(x,y)=x^2e^{-y}+y^2+C

The gradient is continuous for all x,y, so the fundamental theorem of calculus applies, and so the value of the integral, regardless of the path taken, is

\displaystyle\int_C2xe^{-y}\,\mathrm dx+(2y-x^2e^{-y})\,\mathrm dy=f(4,1)-f(1,0)=\frac9e
8 0
3 years ago
I need help finding x!<br><br> Please show work for both!
Alenkinab [10]
1. Answer x=-5
You would subtract 25 from each side of the equal sign, taking 25 away from 40 and 25 away from 25, isolating the x. So you would then have -3x=15. You would then divide each side by -3, and getting x=-5
4 0
3 years ago
Read 2 more answers
HELPPP I DONT GET THISSS
Lina20 [59]
2x + 21 = 35
hope this helps
3 0
3 years ago
Read 2 more answers
What is the image of A(3, 5) under the translation (x - 4, y - 3) ?
azamat
C because (x-4,y-3) = c
4 0
3 years ago
Read 2 more answers
A sporting goods store charges $30 for 12 cans of tennis balls and $45 for two boxes of golf balls. A coach orders100 cans of te
aliya0001 [1]
362.50 for tennis and golf balls.
To find the answer divide the dollar amount by how many cans or boxes each cost then multiply that answer by the total amount purchased.
Golf balls
45/2=22.50 each box
22.5*5=112.50 total
Tennis balls
30/12=2.50 each can
2.50*100 = 250.00
add the dollar amounts  together
112.50+250.00 =362.50
3 0
3 years ago
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