Question

Evaluate I=∫C(sinx+6y)dx+(2x+y)dy for the nonclosed path ABCD in the figure.
A=(0,0),B=(1,1),C=(1,2),D=(0,3)

I=

Answer 1

i got dp/dy=6 and dq/dx=2 so \[\int\limits_{?}^{?}2-6dA\] =\[-4\int\limits_{?}^{?}dA\]

Answer 2

the area of the path is two triangles and a square adding up to 2. -4*2 is what i thought the answer was, but that's wrong. what did i do wrong?

Answer 3

@myininaya can you help?

Answer 4

@perl @e.mccormick .

Answer 5

@phi

Answer 6

Green's theorem doesn't work for nonclosed paths.

Answer 7

oh. so how do i solve this? i have to go to bed soon as i have to be up for work in less than 5 hours. but this is due, so i have to finish first.

Answer 8

what's the figure show?

also, maybe you can "close the figure" , use Green's Thm, then subtract off the path that you added

Answer 9

line integrals confuses me.

Answer 10

You have to parameterize the curve, then evaluate the integral over each segment.

Answer 11

The line segment connecting A and B can be parameterized by
\[C_1:=\begin{cases}x=t\\y=t\end{cases}\text{ for }0\le t\le1\]
The next line would be
\[C_2:=\begin{cases}x=0\\y=2-t\end{cases}\text{ for }0\le t\le1\]
and the final one would be
\[C_3:=\begin{cases}x=1-t\\y=2+t\end{cases}\text{ for }0\le t\le1\]

Answer 12

the line integral from D to A is -9/2
so "correct" your answer by subtracting this off:
-8 - -4.5 = -3.5

Answer 13

Here's the first integral computation as an example.\[\begin{align*}\int_C\cdots&=\int_{C_1}+\int_{C_2}+\int_{C_3}\\\\
&=\int_0^1(\sin t+6t)~dt+(2t+t)~dt+\int_{C_2}+\int_{C_3}\\\\
&=\int_0^1(\sin t+9t)~dt+\int_{C_2}+\int_{C_3}\\\\
&=\left[-\cos t+\frac{9}{2}t^2\right]_0^1+\int_{C_2}+\int_{C_3}
\end{align*}\]

Answer 14

yes, thank you @phi . this whole section is just on green's theorem, so i was wondering why they'd throw this in here if i couldn't use it.

Answer 15

As a check, you can do each individual path, and you should get the same answer.

Answer 16

but the individual paths look a bit messy

Answer 17

yeah. thanks! i'm off to bed now. gotta be awake in 4 and a half hours for work now. thank goodness for naps. lol