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August 7th, 2017, 02:55 PM   #1
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Vector Field upper&lower limit question

Compute
$\displaystyle
\int sin(x) ds
$

sin x ds along the line segment from (-1, 2, 1) to (1, 2, 5).


My solution:
x = -1 + 2t
y = 2
z = 1 + 4t

$\displaystyle
\int_{a}^{b} sin(2t - 1) \sqrt{20} dt
$

How do I determine the upper&lower limit a & b?
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August 7th, 2017, 07:31 PM   #2
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$t=0$ and $t=1$
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August 8th, 2017, 02:10 AM   #3
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Quote:
Originally Posted by romsek View Post
$t=0$ and $t=1$
Is t always within the range of 0 and 1 for any vector field problems?

Last edited by zollen; August 8th, 2017 at 02:22 AM.
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August 8th, 2017, 03:39 AM   #4
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That depends entirely upon how you set up the parametric equations. You have
x = -1 + 2t
y = 2
z = 1 + 4t
as the line from (-1, 2, 1) to (1, 2, 5).

x= -1+ 2t= -1 so 2t= 0, t= 0. y= 2 for all t. z= 1+ 4t= 1 so 4t= 0, t= 0 again.
x= -1+ 2t= 1 so 2t= 2, t= 1. y= 2 for all t. z= 1+ 4t= 5 so 4t= 4, t= 1 again.

Please, please, please, learn why a formula works rather than just memorizing! I suspect that you learned that parametric equations for the line from $\displaystyle (x_0, y_0, z_0)$ to $\displaystyle (x_1, y_1, z_1)$ are given by $\displaystyle x= x_0+ (x_1- x_0)t$, $\displaystyle y= y_0+ (y_1- y_0)t$, and $\displaystyle z= z_0+ (z_1- z_0)t$. Do you see why that works? If t= 0 then we have only $\displaystyle x_0$, $\displaystyle y_0$, and $\displaystyle z_0$. When t= 1 we have $\displaystyle x_0+ x_1- x_0= x_1$, $\displaystyle y_0+ y_1- y_0= y_1$, and $\displaystyle z_0+ z_1- z_0= z_1$.

But you don't need to memorize a formula. We know that, since this is a straight line, we can take the parametric equations to be "linear": x= at+ b for some a and b. We choose to take t= 0 at the first point, (-1, 2, 1) and t= 1 at the second point, (1, 2, 5). So when t= 0, we want x= -1= a(0)+ b, b= -1. When t= 1, we want 1= a(1)+ b= a- 1 so a= 2: x= 2t- 1. Similarly, taking y= at+ b, when t= 0, we want y= 2= a(0)+ b so b= 2. When t= 1, we want y= 2= a(1)+ 2 so a= 0: y= 2. Finally, taking z= at+ b, when t= 0, we want z= 1 so z= 1= a(0)+ b so b= 1. When z= 1, z= 5= a(1)+ 1 so a= 4: z= 4t+ 1.

But we could as well have chosen the parameter, t, to go from, say, 1 to 2:
Since this is a line, we can take the parametric equation to be linear, x= at+ b. When t= 1, I want x= -1 so I want -1= a+ b. When t= 2, I want x= 2 so 2= 2a+ b. Subtracting a+ b= -1 from 2a+ b= 2, a= 3. Then 3+ b= -1 so b= -4: x= 3a- 4. Similarly we get y= 2 and z= 4t- 3. Those give, for t= 1, (-1, 2, 1) and, for t= 2, (1, 2, 5) as before. It's just much simpler to take the endpoints to be at t= 0 and t= 1.

While parametric equations for a straight line can be linear, they don't have to be. Since there are two points we just need two coefficients in each equation. We could, for example, take x= a t^3+ b with t from 1 to 2. Then we must have x= -1= a(1)+ b and x= 1= a( 8 )+ b. Subtracting a+ b= -1 from 8a+ b= 1, we have 7a= 2 so a= 2/7. Then a+ b= 2/7+ b= -1 so b= -1- 2/7= -9/7. x= (2/7)t^3- 9/7 and similarly for y and z.

Of course, there is no good reason to do all that extra work when we can take the equations to be linear and the t values at the endpoints the simple numbers t= 0 and t= 1.
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August 8th, 2017, 05:29 AM   #5
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Quote:
Originally Posted by zollen View Post
Is t always within the range of 0 and 1 for any vector field problems?
it all depends on how you parameterize it.

I tend to use parameterizations that range from $t=0$ to $t=1$ but you don't have to.

It looks like Joppy expounds upon this at length.
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