Stuck on line integral: F=⟨y+z,z+x,x+y), γ is 1/4-circle w/ center (1,1,1)

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Cactusguy

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Stuck on line integral: F=⟨y+z,z+x,x+y), γ is 1/4-circle w/ center (1,1,1)

Hi, I'm having trouble with a line integral question in my calculus homework.

Here is the question:

Evaluate the following line integrals ∫γ F⋅ds where
a) F=⟨y+z,z+x,x+y⟩and γ is the quarter circle with center (1,1,1) going from (1,0,1) to (0,1,1).
b) F=⟨y4,x3⟩ and γ is the path along the graph of y=x2 from x=-2 to x=1.


I'm just going to focus on a) for right now to make it easier.
I know the formula is to take the integral of F times ds

To find the quarter circle, I parametrized it so that
z=1
x=1+cos(t)
y=1+sin(t)

Thus, the lower bound is 3pi/2 and the upper bound is pi/2, and that draws the correct path because it goes clockwise around the circle.

thus

dz/dt = 0
dy/dt=cos(t)
dx/dt -sin(t)

ds= sqrt( cos^2 (t) + sin^2 (t))
ds = sqrt(1)
ds = 1

But now what?

The actual F given is weird, it's a vector or something.
F=⟨y+z,z+x,x+y⟩
What am I supposed to do with that? If it was a regular equation I'd just substitute my parameterized values for x y and z and integrate but I don't know what to do with a vector like that.

Can anyone help me?

Thanks.
 
1) parameterize the curve as you've done, I think you'll find the limits are \(\displaystyle \dfrac {3\pi}{2} \to \pi \)

2) using that parameterization, call it \(\displaystyle \vec{r}(t)\), find \(\displaystyle d\vec{r} = \dfrac{d}{dt} \vec{r}(t) \)

3) using \(\displaystyle \vec{r}(t)\) rewrite \(\displaystyle F(x,y,z) \text{ as }F(x(t),y(t),z(t))\)

4) find \(\displaystyle F(x(t),y(t),z(t))\cdot d\vec{r}(t)\)

5) integrate \(\displaystyle \displaystyle \int_{\frac{3\pi}{2}}^{\pi} ~F(x(t),y(t),z(t))\cdot d\vec{r}(t)~dt \)
 
Romsek said:
1) parameterize the curve as you've done, I think you'll find the limits are \(\displaystyle \dfrac {3\pi}{2} \to \pi \)

2) using that parameterization, call it \(\displaystyle \vec{r}(t)\), find \(\displaystyle d\vec{r} = \dfrac{d}{dt} \vec{r}(t) \)

3) using \(\displaystyle \vec{r}(t)\) rewrite \(\displaystyle F(x,y,z) \text{ as }F(x(t),y(t),z(t))\)

4) find \(\displaystyle F(x(t),y(t),z(t))\cdot d\vec{r}(t)\)

5) integrate \(\displaystyle \displaystyle \int_{\frac{3\pi}{2}}^{\pi} ~F(x(t),y(t),z(t))\cdot d\vec{r}(t)~dt \)
Click to expand...

See, that's my issue though. How on earth do I integrate a vector?
 
I am surprised that you have reached problems like this without learning that \(\displaystyle \int \begin{pmatrix}f(t) \\ g(t) \\ h(t)\end{pmatrix}dt= \begin{pmatrix}\int f(t)dt \\ \int g(t)dt \\ \int h(t)dt \end{pmatrix}\)

In other words, integration of vectors is "coordinate wise".
 
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Thanks for the help guys and gals. I was just having a total brain fart, I got it figured out. My question I asked was wrong.
 
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